# Institute for Mathematical Sciences Programs & Activities

**Representation Theory of Lie Groups**

(July 2002 - January
2003)

(July 2002 - January 2003)

**~ Abstracts ~**

Organizers · Confirmed visitors · Overview · Schedule of talks and tutorials · Papers and lecture notes

**Character theory of non-linear
real groups**

*Jeffrey Adams, University of Maryland, College Park, USA*

This is joint work with Peter Trapa and Rebecca Herb, and related to the talks of David Renard. I will discuss two closely related phenomena: Vogan duality and character lifting (along the lines of Shelstad's theory) for non-linear real groups. The case of the metaplectic group Mp(2n,R) and the two-fold cover of GL(n) are related to the theta-correspondence and Kazhdan-Patterson lifting respectively. A general theory, at least in the case of one root length, is available.

**unitarity@home**

*Jeffrey Adams, University of Maryland, College Park, USA*

The unitary dual of a real Lie group is only known in some special cases. For any given Lie group there is a finite algorithm to compute the unitary dual. I will discuss a proposal to compute the unitary dual of real Lie group by computer.

**Representations of p-adic Groups**

*Gordan Savin, University of Utah, USA*

This set of tutorials provide an elementary introduction, with exercises, to representations of p-adic reductive groups.

We start with definitions and preliminary results on p-adic fields, structure of GL_n(F) over a p-adic field F, and smooth representations. In order to keep the exposition as simple as possible, we restrict ourselves to GL_2(F). However, the topics and their proofs are chosen so that they easily generalize to GL_n(F) and other reductive groups.

Next, we introduce induced and cuspidal representations, and prove that irreducible smooth representations are admissible. We shall also discuss the composition factors of induced representations, unitarizable representations, and construct the complementary series for GL_2(F).

Finally, we go back to GL_n(F) and describe the composition factors of (regular) induced representations. This result, due to Rodier, is based on the combinatorics of the root system, and the reduction to the special case of GL_2(F). As such, it gives a good introduction to further, more advanced topics.

**Dirac operators in
representation theory**

*Jing Song Huang, Hong Kong University of Science and
Technology, Hong Kong and Pavle Pandzic, University of
Zagreb, Croatia*

Dirac operators are widely used in the index theory in differential geometry and geometric construction of discrete series representations. The aim of this tutorial is to reveal the algebraic nature of Dirac operators, namely Vogan's conjecture on Dirac cohomology. We will explain a proof of this conjecture and show its wide applications in representation theory. We will also include some background material on Lie groups representations and Lie algebra cohomology.

The 10 hour lectures are tentatively arranged as follows:

- (g,K)-modules
- Clifford algebra & Spinors
- Dirac operators & group representations
- Aq(lambda)-modules
- Vogan's Conjecture and its proof
- Borel-Weil Theorem and Discrete Series
- Lie algebra cohomology
- (g,K)-cohomology
- Dirac cohomology and other cohomologies
- Multiplicity of automorphic forms

**Orbits of Lie Groups**

*Hung-Yean Loke, National University of Singapore,
Singapore*

We will cover the basic definitions and classifications of semisimple and nilpotent orbits. If time permits, we will describe in detail some interesting exceptional orbits and their coordinate rings.

**Degenerate Principal Series
Representations of GL(n,C)**

*Soo-Teck Lee, National University of Singapore,
Singapore*

Let G=GL(n,C) and P the parabolic subgroup of G with its Levi factor isomorphic to GL(n-k,C)xGL(k,C). We consider the representation of G induced from a character of P. We shall calculate explicitly the action of the Lie algebra of G on this representation. This allows us to determine its reduciblitly, composition series and unitarity.

**Uncertainty and Entropy in
Time-Frequency: Finite versus Continuous**

*Tomasz Przebinda, University of Oklahoma, Norman, USA*

In 1957 Hirschman proved that the sum of entropies of a
function *f* (with )
and its Fourier transform is nonnegative, [Hi]. He also
observed that a stronger version of this inequality:

(proven later by Beckner [B, page 177]) implies the Heisenberg-Weyl Uncertainty Principle. Hirschman conjectured that the minimizers for the sharp inequality (1) were Gaussians, as is the case for the Heisenberg-Weyl Uncertainty Principle. We shall show that this is indeed the case.

There is an analog of (1) for functions * f* defined on a finite abelian group, with applications in
Signal Processing. The shall also describe these
minimizers. They depend on the structure of the finite
abelian group and are not "Gaussians" or discretized
Gaussians. This discrepancy between the finite and the
continuous case seems to be unexpected in the Signal
Processing community. In fact the main motivation for
this work is to expose this fundamental difference.

We shall spend considerable amount of time explaining the notions involved, including the entropy and the representation theory of the groups we shall need.

References:

[B] W. Beckner, *Inequalities in Fourier Analysis*,
Annals of Mathematics 102 Number 6 (1975), 159 - 182.

[Hi] I. I. Hirschman, Jr., *A Note on Entropy*, Amer.
J. Math. 79 (1957), 152 - 156.

**An explicit trace formula**

*Benedict Gross, Harvard University, USA*

In this talk, we will illustrate some recent progress that has been made on the trace formula, by Arthur and Kottwitz. We will give a formula for the Euler characteristic of the cohomology of the discrete spectrum. This integer is given as the sum of rational numbers, indexed by stable torsion conjugacy classes. The main contribution of each class is a product of certain values of Artin L-functions, at negative integers.

**A comparison of Haar measures**

*Benedict Gross, Harvard University, USA*

A locally compact topological group G has a
left-invariant measure d> unique up to scalar multiples.
For G compact, we can normalize the measure so that .
For G discrete, we can normalize so that * >g) = 1*, for all g in G.

In certain situations, which we will consider, there are two naturally defined invariant measures on G, and we will want to determine their ratio. For example, if G is finite, it is both compact and discrete. The ratio of the measures given above is just the order of G. We will consider the situation when G is the group of points of a reductive algebraic group over a finite or local field, and when G is the group of adelic points of a reductive algebraic group over a global field. The ratios are computed using L-functions associated to G.

**Mappings which preserve
familes of curves**

*Michael Cowling, University of New South Wales,
Australia*

This talk is a survey of some results in geometry from Darboux to Tits. First, any bijection of the plane which maps lines to lines is affine, i.e., a composition of a linear map and a translation. There are many local versions of this result: on one hand, a bijection of the open unit square which preserves line segments is linear, while on the other, there are nonlinear (but fractional linear) maps of the open unit circle preserving chords.

Next, we consider the plane and the sphere. All bijections of the projective plane which preserve projective lines come from the projective group, while there are many maps of the sphere which preserve great circles which are not even continuous.

Finally, we consider maps of three-dimensional space which preserve two families of lines, one family of lines parallel to the y axis and the other family of lines lying in planes parallel to the xz plane and with gradients equal to the y coordinates. It is shown that these maps are affine, and that this implies (for SL(3,R)) a theorem of Tits that the morphisms of a spherical building come from the group. A local version of this result is also outlined.

**The Geometry of K/M**

*Michael Cowling, University of New South Wales,
Australia*

The analytic/geometric construction of complementary series realizes these exceptional unitary representations on spaces of Sobolev type. Indeed, in the classical case of SO(1,n), the kernels of the intertwining operators may be expressed in terms of the euclidean distance and the corresponding representations may be represented on Sobolev spaces defined using the classical laplacian. In the general rank one case, an analytic description of these spaces relies on a noneuclidean distance and a sublaplacian. In the rank one case, these fit into the not-yet-mature theory of Carnot-Caratheodory manifolds.

In the higher rank case, it is less clear what is the appropriate geometry for K/M, or more generally G/P where P is a parabolic subgroup. This talk, based on joint work with Filippo De Mari, Adam Koranyi and Hans Martin Reimann, is about this problem. What is the notion of conformal geometry appropriate to the higher rank case? We give elementary proofs of some results of Yamaguchi on this question.

**Homomorphisms of the
icosahedral group into reductive groups**

*George Lusztig, MIT, USA*

In the representation theory theory of finite groups one studies homomorphisms from a finite group into GL_n up to conjugacy. More generally GL_n could be replaced by any reductive group. We are mainly interested in the case where the finite group is the icosahedral group. The most interesting case is that where the reductive group is of type E_8. In this case we find the complete classification of homomorphisms up to conjugacy completing earlier work of D.D.Frey.

**K-types and character expansions**

*Julee Kim, University of Michigan, Ann Arbor, USA*

Let k be a p-adic field, and let G be a group of k-points of a connected reductive group defined over k. For an irreducible admissible representation of G, we discuss an asymptotic expansion of its character. This expansion depends on K-types contained in the given representation. This is a joint work with Fiona Murnaghan.

**Dual blobs and Plancherel
formulas**

*Julee Kim, University of Michigan, Ann Arbor, USA*

Continuing from the previous talk, we discuss some applications of character expansions.

**Theta lifting of nilpotent
orbits for symmetric pairs**

*Chen-Bo Zhu, National University of Singapore, Singapore*

We consider a reductive dual pair (G, G') in the stable range with G' the smaller member. We study theta lifting of nilpotent K'_C -orbits, where K' is a maximal compact subgroup of G, and describe the K_C module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair (G, K). Connection with the K-type structure of theta lift of unitary representations of G' will also be discussed. This is joint work with Kyo Nishiyama of Kyoto University.

**Signatures of invariant
Hermitian forms: general theory**

*David Vogan, MIT, USA*

Let G be a real reductive Lie group, K a maximal compact subgroup, and X an irreducible Harish-Chandra module for G. By a theorem of Harish-Chandra, X comes from a unitary representation of G if and only if X admits an invariant Hermitian form that is positive definite.

By a theorem of Knapp, it is easy to tell when X admits an invariant Hermitian form. I will explain how to define a "signature" for such an invariant form, and what such signatures can look like in general. One can then formulate the problem of computing the signature of any invariant Hermitian form. This problem includes the problem of determining the unitary dual of G. I'll explain a few results and many open problems about this computation.

**Computing signatures of
invariant Hermitian forms**

*David Vogan, MIT, USA*

Let G be a reductive group over a local field k, and P = LN a minimal parabolic subgroup of G. Suppose K is a compact subgroup of G with the property that G = KP and the intersection of K with P is a subgroup M of L. (This allows us to identify the homogeneous spaces G/P and K/M.) Write a^* for the (real vector space) of characters of L taking positive real values; such characters are trivial on M.

Given a character nu in a^*, we extend nu to a character of P by making the unipotent radical N act trivially, and then form the principal series representation I(nu) = Ind_P^G (nu). These representations are defined on a common Hilbert space H = L^2(K/M).

Except in case nu=0, the representation I(nu) is not unitary with respect to the natural Hilbert space structure on H. However, one can sometimes find a nice family <,>_nu of hermitian forms on H, with the property that I(nu) preserves the form <,>_nu. If the form <,>_nu is semi-definite, one finds in this way a unitary composition factors of I(nu); these are "spherical complementary series" representations.

One would therefore like to know exactly when the forms <,>_nu are semi-definite. When G is split and k is a non-archimedean field, Barbasch and Moy have shown that the semi-definiteness of <,>_nu can be made into a question in the group algebra of the Weyl group. One beautiful and immediate consequence is that the answer does not depend on the field k.

I will explain the result of Barbasch and Moy. Part of their condition is that a certain matrix depending on nu (of size equal to the dimension of a^*) be positive semidefinite. The rank of this matrix was computed by Joseph some fifteen years ago; now one wants to know whether its non-zero eigenvalues are all positive.

Finally, I will discuss work with Salamanca and Barbasch providing consequences of these calculations for the archimedean fields R and C.

**Harmonic analysis for p-adic
groups**

*Stephen Debacker, University of Chicago, USA*

This set of tutorials will attempt to provide an elementary introduction to harmonic analysis on p-adic groups. This area of mathematics is very technically demanding. Thus, in order to keep the primary focus on the ideas rather than the details, we will restrict our attention to the general linear group GL_n, and, when required for expository reasons, we will further restrict our attention to the group GL_2. Everything we discuss is valid in a far more general context.

We start by introducing the basic objects: orbital integrals, characters of irreducible admissible representations, and Fourier transforms (on the Lie algebra of our group). The ultimate goal is to understand the Harish-Chandra--Howe local character expansion which provides a very beautiful connection among these objects.

We next restrict our attention to nilpotent orbital integrals. In this context, we discuss a fundamental result of Deligne and Ranga-Rao which states that orbital integrals define invariant distributions, and we will then turn our attention to Huntsinger's proof that the Fourier transform of an orbital integral is represented by a locally integrable function on the regular set.

Next we discuss a version of Howe's conjecture for the Lie algebra. Note: This "conjecture" has been a theorem for well over 25 years.

Finally, we pull everything together by discussing the Harish-Chandra--Howe local character expansion. This is a very deep result which states that on some neighborhood of the identity, the character of an irreducible addmissible representation can be written as a linear combination of the Fourier transforms of nilpotent orbital integrals.

**Branching laws for p-adic
groups**

*Dipendra Prasad, Mehta Research Institute, India*

In these lectures we will introduce the concept of sphercal pairs, i.e., pair of groups (G,H) with G reductive such that the action of H on the flag variety of G/B has an open orbit. In this situation, finite dimensional representations of the algebraic group G have at most 1 dimensional space of H-invariant vectors. We will look at some examples, and see what happens for infinite dimensional representations of p-adic groups. We will try to give some examples, and some proofs.

**Invariant differential operators
on a classical lie supergroup**

*Tomasz Przebinda, University of Oklahoma, Norman, USA*

An invariant eigendistribution on a real reductive group is a distribution which is in- variant under the conjugation by elements of the group, and is an eigendistribution for the commutative algebra of left and right invariant differential operators on the group. A real reductive dual pair, together with the underlying symplectic space, may be viewed as a classical Lie supergroup. We shall show how does the notion of an invariant eigendistribution extend to this context. In particular we shall recover the correspondence of infinitesimal characters for representations under Howe's correspondence.

**Invariant eigendistributions on
a classical lie supergroup**

*Tomasz Przebinda, University of Oklahoma, Norman, USA*

The goal of this lecture is to explain the Harish-Chandra's method of descent in the context of a classical Lie supergroup and to apply it to study the characters of representa- tions which occur in Howe's correspondence. We shall also explain a conjectural relation between these characters, which may be traced back to the Cauchy determinant identity.

**Degenerate Principal Series
Representations of U(p,q)**

*Soo-Teck Lee, National University of Singapore,
Singapore*

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**Generic representations of
reductive groups over finite rings**

*George Lusztig, MIT, USA*

Abstract. Let G be a connected reductive algebraic group defined over a finite field F_q. Let r be a strictly positive integer. We are concerned with the study of complex representations of the finite group G(F_q[[Z]]/(Z^r)) where Z is an indeterminate. Using a cohomological method (etale cohomology) extending that of Deligne and the author (1976), we construct an irreducible representation of G(F_q[[Z]]/(Z^r)) for any "maximal torus" and a generic character of it; for r at least 2, this was stated without proof in a paper I wrote in 1977.

**Group schemes associated to
Moy-Prasad groups**

*Jiu-Kang Yu, University of Maryland, USA*

We construct canonical smooth integral models of p-adic reductive groups associated to groups in the theory of Moy-Prasad and Schneider-Stuhler, and study their properties.

**What are minimal
representations of p-adic groups?**

*Gordan Savin, University of Utah, USA*

In this talk we shall compare several definitions of minimal representations over p-adic fields. Next, we shall describe an approach to minimal representations due to Weissman, and discuss how its properties can be used to establish results on dual pair correspondences.

**Commuting differential
operators with regular singularities at infinity and
completely integrable quantum systems**

*Toshio Oshima, University of Tokyo, Japan*

In this talk we discuss commuting differential operators defined at an infinite point of a Cartan subgroup. They include Heckman-Opdam hypergeometric systems, differential equations satisfied by Whittaker vectors, finite Toda chains etc. We give their classification and their explicit expression.

**Cycle Spaces and Representations**

*Joseph Wolf, University of California at Berkeley, USA*

I'll recall some geometric constructions of tempered representations of real reductive Lie groups, and some possibilities for going to non-tempered representations. The latter will concentrate on double fibration transform methods, e.g. the complex Penrose transform, and the role of the linear cycle space of a flag domain.

**Structure of the Cycle Spaces**

*Joseph Wolf, University of California at Berkeley, USA*

I'll describe some very recent results that give the precise structure of the linear cycle space of a flag domain. Roughly speaking, all the famous tubular neighborhoods of G/K, in its complexification, coincide; and they are equal to the cycle space except in certain specific cases. The implications for double fibration transforms will be discussed.

**Supercuspidal character germs
of classical and other groups**

*Jeffrey Adler, University of Akron, USA*

Supercuspidal character germs ought to be expressible as an explicit linear combination of Fourier transforms of elliptic orbital integrals. In most cases, only one orbit should be involved. We will show how to prove this result for many supercuspidal characters of many groups.

**Base change, with an example**

*Jeffrey Adler, University of Akron, USA*

Given some knowledge of characters and of Bruhat-Tits theory, it is possible to make base change more explicit in certain cases. We look at an example involving base change from U(3) to GL(3). This is joint work in progress with Joshua Lansky.

**On self-dual representations of
p-adic groups**

*Dipendra Prasad, Mehta Research Institute, India*

An irreducible self-dual representation of a group carries an invariant bilinear form which is unique up to scaling. It is therefore either symmetric or skew-symmetric. It is a classical theorem for compact Lie groups that the bilinear form is symmetric or skew-symmetric depending on the action of an element in the centre of the group of order less than or equal to 2. In this talk we look at the analogue of this theorem for p-adic groups. We summarise the results for generic representations of quasi-split groups which appeared in IMRN (1999) which gives a rather close analogue for many groups, including groups with trivial centre, and classical groups such as GL(n), Sp(n), SO(n), but works only for SL(n) when n is not congruent to 2 modulo 4. There are counter-examples for such an expectation for SL(6).

We formulate a conjecture for general quasi-split group in terms of lifting of the corresponding Langlands parameter to a certain 2 fold cover of the L-group.

As a specific non-quasisplit group, we discuss the case for division algebras, and state the conj. made with Dinakar Ramakrishnan which relates the parity of the bilinear form of an irreducible self-dual representation of the invertible elements of a division algebra in terms of the corresponding information on the Galois representation.

**Tensor Product of Degenerate
Series and Theta Correspondence**

*Eng-Chye Tan, National University of Singapore,
Singapore*

We shall describe an intertwining map from the oscillator representation to the tensor product of two degenerate principal series representations of G and G' (which form a reductive dual pair in the sense of Howe). We shall give examples of results on local theta correspondence for some pairs.

**Equidistribution of Cycles on
Hilbert Modular Varieties**

*Jian-Shu Li, Hong Kong University of Science and
Technology, Hong Kong*

The Andre-Oort conjecture is an assertion on the Zariski density of CM-points on Shimura varieties. This can be viewed as an equal distribution property of zero dimensional Shimura subvarieties. In this talk we shall discuss equal distribution of subvarieties on Hilbert modular varieties defined by quaternion algebras. This is a report of joint work in progress with D. Jiang and S. Zhang.

**A Vanishing Result for the
Cohomology of Arithmetic Groups**

*Jian-Shu Li, Hong Kong University of Science and
Technology, Hong Kong*

I will outline a proof of the following vanishing theorem (joint work with Schwermer): Let $G$ be a semi-simple Lie group and $K$ a maximal compact subgroup. Let $$ q_0(G) = \frac{1}{2}[\dim G/K - rank(G) + rank(K)] $$ Suppose $E$ is a finite dimensional representation of $G$ with regular infinitesimal character and $\Gamma$ is an arithmetic subgroup of $G$. Then $H^j(\Gamma, E)=0 $ for all $j<q_0(G)$.

**Homogeneity Results for
Reductive p-adic Groups**

*Stephen Debacker, University of Chicago, USA*

In the 1970s Roger Howe made two "finiteness" conjectures concerning invariant distributions on reductive p-adic groups and their Lie algebras. These conjectures were answered in the affirmative for the Lie algebra by Harish-Chandra and Howe and for the group by Clozel (in characteristic zero) and Barbasch and Moy (in general). In the 1990s Waldspurger proved a very precise version of the Howe conjecture for the Lie algebra for classical unramified groups. In this talk, we will discuss a generalization of Waldspurger's result and discuss possible applications.

**Quantum Analogues of
Coherent Families at Roots of 1**

*R. Parthasarathy, School of Mathematics, Tata Institute
of Fundamental Research*

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**Theta lifting, affine
quotients, and degree formula for highest weight modules
I.**

*Kyo Nishiyama, Kyoto University, Japan*

We begin with a compact dual pair, and give the Bernstein degree of unitary highest weight modules using a theorem of invariant theory. After that, we will show how to generalize the idea to non-compact dual pair. We explain how to lift nilpotent orbits (or associated variety), and realize them as an affine quotient of a bigger nilpotent orbit which is lifted from the trivial orbit. This talk is based on a joint work with Chengbo Zhu (NUS).

**Theta lifting, affine
quotients, and degree formula for highest weight modules
II.**

*Kyo Nishiyama, Kyoto University, Japan*

Let (G, G') be a dual pair of type I. In the stable range, we give a formula of lifting of nilpotent orbits. There are many interesting examples of lifted nilpotent orbits, and we explain some application to branching rules and multiplicity free actions. Finally, we give an exact formula of the associated cycle of the theta lifting of unitary highest weight modules in the stable range. This talk is based on a joint work with Chengbo Zhu (NUS).

**Dual pair PGL(3) x G_2 in E_6,
and (g_2, SL(3))-modules**

*Gordan Savin, University of Utah, USA*

Let G_2 be the exceptional complex Lie group, and g_2 its Lie algebra. Since SL(3) is a spherical subgroup of G_2, the theory of (g_2, SL(3))-modules is a good one. In this lecture we describe a classification of irreducible (g_2, SL(3))-modules, and construct a correspondence between (algebraic) representations of complex PGL(3) and (g_2, SL(3))-modules using a "minimal representation" of E_6. On the level of infinitesimal characters the correspondence is functorial for the inclusion of dual groups SL(3) --> G_2. This then give us Capelli-type identities in the eveloping algebra of E_6, modulo the Joseph ideal, such as C + 2=3C' + 14, where C and C' are the Casimir operators of PGL(3) and G_2, respectively.

**On embeddings of derived
functor modules into degenerate principal series I & II**

*Hisayosi Matumoto, University of Tokyo, Japan*

The definition of the Sato hyperfunction suggests existence of embeddings of derived functor modules into degenerate principal seires as boundary value maps. However, it seems difficult to construct an embedding of a global cohomolgy such as a derived functor module via a local method. However, in some case, we can show existence (or nonexistence) of such a hoped-for embedding of a derived functor module via another method. In this talk, I would like to explain, in particular, the complex group case. In this case, existence of a homomorphism between generalized Verma modules is seriously related to existence of a embeddings of a defived fuctor module.

In part II, I will discuss examples of embedding of derived functor modules into degenerate principal series.

**Isotropy representations
for singular unitary highest weight modules**

*Hiroshi Yamashita, Hokkaido University, Japan*

We describe the isotropy representation ${\mathcal W}_\lambda$ attached to every singular unitary highest weight module $L(\lambda)$. In the oscillator setting, it has been already shown that the assignment ${\mathcal W}_\lambda^\ast \leftrightarrow L(\lambda)$ essentially gives the Howe duality correspondence with respect to a compact dual pair. In this talk, We focus our attention on $L(\lambda)$'s which can not be realized by the theta correspondence. By using the projection onto the PRV-component, the isotropy representations are explicitly determined for such highest weight modules. This gives in particular a clear understanding of the multiplicity formulae obtained by Kato and Ochiai for the cases BI, DI and EVII. Moreover, it turns out that the representation ${\mathcal W}_\lambda$ is irreducible for every singular unitary highest weight module. This is a joint work with Akihito Wachi of Hokkaido Institute of Technology.

**Eigenspace representations of
symmetric spaces of exceptional type**

*Hiroyuki Ochiai, Tokyo Institute of Technology, Japan*

For a classical, complex, or rank-one symmetric space, every invariant differential operator comes from the center of the corresponding universal enveloping algebra. The classification of the symmetric pairs which do not have this property is completed by Helgason in 90s with some additional information. In this talk, I'll discuss a more detailed structure of the image of the center of the enveloping algebra inside the ring of invariant differential operators and the structure of the eigenspace representations. I'll also mention the relation with the work by Huang-Oshima-Wallach on the generalized eigenfunctions on such symmetric spaces.

**Theta correspondence,
one-dimensional representations, and unitarity**

*Annegret Paul, Western Michigan University, USA*

I will discuss the theta correspondence of unitary one-dimensional representations, starting with the case of the dual pairs (U(p,q),U(r,s)) (this is joint work with Peter Trapa). In particular, I'll talk about the questions of occurrence, unitarity and Langlands parameters of the theta lifts, and which families of unitary (unipotent?) representations occur.

**On Finite Subgroups of Some
Linear Groups**

*Sun Binyong, Hong Kong University of Science and
Technology*

The so called torsion conjecture says: Given any abelian variety A defined over a number field k, the order of the torsion part of A(k) is bounded by a constant C(k, d) which only depends on the number field k and the dimension d of the abelian variety. If we replace the abelian variety by certain linear groups, the analogue is much easier. We get some results on this direction.

**Local Langlands conjecture for
nonlinear real Lie groups**

*Peter E. Trapa, University of Utah, USA*

Let G~ be a nonlinear double cover of a linear reductive real Lie group G. The purpose of this talk is to establish some connections between the representation theory of G and representation theory of an associated linear real group H. (H turns out to be a real form of a subgroup of the Langlands dual of the complexification of G.) This includes a matching of parameters for representations of G~ and H which is compatible, in an appropriate sense, with stability. For instance, when G = Sp(2n,R), we find that H = O(p,q). In this case, the matching of parameters is closely related to the theta correspondence, and the appropriate stabilization recovers a lifting of Adams. As a further example, when G = GL(n,R), we obtain a version of Kazhdan-Patterson lifting. This is joint work with Jeffrey Adams and David Renard. This talk is an overview; later talks of Renard will provide significantly more details and results.

**New restrictions on
characteristic cycles of Harish-Chandra modules**

*Peter E. Trapa, University of Utah, USA*

We discuss various structures -- some geometric, some algebraic -- on the computation of characteristic cycles of Harish-Chandra modules. This leads to many new computations. For instance, we find families of representations whose characteristic cycles have (roughly) 2^{r/2} irreducible components, where r is the split rank. The calculations suggest a kind of duality for the leading piece of the characteristic cycle.

**K-type Structure of Degenerate
Principal Series**

*Roger Howe, Yale University, USA*

This talk will show in several examples, how the analysis by Howe and Lee, of the most degenerate principles series for GL_n (R) and GL_n (C), can be used to find interesting information about more general principal series. This is in part joint work with S-T Lee, and part a report on calculations by C. Will.

**Small unitary representations**

*Siddhartha Sahi, Rutgers University, USA*

We describe explicit models of small unitary representations of certain semisimple Lie groups and establish a theta-type correspondence arising from their tensor products.

**Deformation quantization and
invariant distributions**

*Siddhartha Sahi, Rutgers University, USA*

The exponential map carries invariant germs of distributions on the Lie algebra to those on the Lie group. An conjecture of Kashiwara-Vergne asserts that after a certain universal twist (motivated by the orbit method) this is actually an isomorphism for the (partial) algebra structures on the two spaces. We describe a proof of this result based on Kontsevich's star-product construction.

**Representations with Scalar
K-types and Applications**

*Chen-Bo Zhu, National University of Singapore, Singapore*

We discuss some results of Shimura on invariant differential operators and extend a folklore theorem about spherical representations to representations with scalar K-types. We then apply the result to obtain non-trivial isomorphisms of certain representations arising from local theta correspondence, many of which are unipotent in the sense of Vogan.

**Conformal geometry and
analysis on minimal representations of O(p,q)**

*Toshiyuki Kobayashi, RIMS, Kyoto, Japan*

We apply methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite dimensional representation of the conformal group on the solution space of the Yamabe equation. Then, I will discuss various geometric models together with explicit inner products of the minimal representation of O(p,q).

**Parameterizations via
Bruhat-Tits Theory**

Stephen Debacker, University of Chicago, USA

We shall discuss various parameterizations which arise naturally from Bruhat-Tits theory.

**Cohomology of Compact
Quotients of Hermitian Domains and the Role of the
Modules $ A_{q,}$**

*R. Parthasarathy, School of Mathematics, Tata Institute
of Fundamental Research*

Some irreducible unitary representations contribute to the cohomology of compact quotients of symmetric spaces via 'Matsushima's formula'. We recall them and describe some known facts about them. Towards the end, we will briefly discuss some results of T.N. Venkataramana on restriction of cohomology classes to some submanifolds arising from reductive subgroups and secondly some results about the Hodge decomposition (for the hermitian case).

**Bernstein center distributions**

*Marko Tadic University of Zagreb, Croatia*

Bernstein center of a reductive $p$-adic group is an analogue of the center of the enveloping algebra of a Lie algebra. It has a number of very important applications. The center is usually studied as regular functions on the algebraic variety of (infinitely many) connected components. It is isomorphic to the space of invariant distributions which have essentially compact support (this means that these distributions are compactly supported after convolution with any locally constant compactly supported function). The isomorphism is given by "Fourier transform".

The delta distribution supported at identity is in Bernstein center. It is not so obvious to point out some other elements and a question is how to describe more explicitly this huge family of distributions. This is topic of our talk. The results that we shall present are obtained jointly with A. Moy (they are part of a longer joint project).

**Macdonald Positivity
Conjecture, n! Conjecture, G-Hilbert Scheme and Diagonal
Harmonics**

*Claudio Procesi, Universita' di Roma, Italy*

In 89 Macdonald discovered a remarkable class of symmetric polynomials depending on 2 parameters and conjectured that, once expanded in terms of Schur polynomials, they have coefficients which are polynomials in the two parameters with positive integer coefficients. Garsia and Haiman developed a strategy to prove this conjecture using a certain bigraded representation of the symmetric group, finally this led to the n! conjecture which has been recentely solved by Haiman using deep properties of the Hilbert scheme. The same techiques have allowed to solve conjectures on diagonal harmonics.

**The associated variety of a
unipotent representation**

*Dan Barbasch, Cornell University, USA*

The associated cycle of a Harish-Chandra module is a linear combination with integer coefficients of nilpotent orbits. The associated variety is the (closure of the) union of the orbits which occur with nonzero multiplicity. One use of the associated cycle is via results of Adams-Barbasch-Vogan which show how to construct stable (in the sense of Langlands) combinations of characters. Another use is to derive information about composition series of induced representations.

In this talk I will describe the cycle of the special unipotent representations in the case of the classical real algebras sp(n) and so(p,q). As a consequence one obtains a description of the associated variety of each Harish-Chandra cell.

**The geometry of admissible
representations**

*Jing Song Huang, Hong Kong University of Science and
Technology, Hong Kong*

In this talk we would like to show the connection between the moment map for the associated variety and the condition being admissible for group representations.

**Branching of Minimal
Holomorphic Representations**

*Zhang Genkai, University of Chalmers, Sweden*

We study the tensor product decomposition of a minimal holomorphic representation with its complext conjugate. It has continuous part and in certain cases also a descrete parts consisting of complementary series. We find some (new) unitary spherical representations. We compute the Clebsch-Gordan coefficients and study the quantization of the complementary series representations.

**The Structure of the Ring of
Quasi-symmetric Polynomials**

*Nolan Wallach, University of California at San Diego,
USA*

The ring of quasi-symmetric polynomials n-variables is a subring, QS, of the polynomials that contains the symmetric polynomials, S. In this lecture I will explain the ingredients of a proof (due to the speaker and A.Garsia) of a conjecture of Bergeron and Reutenauser that says that QS is free as an S-module. This implies, in particular, that QS is a Cohen-Macaulay ring. The proof involves a very interesting element of the group algebra of the symmetric group which has played an important role in the speakeræŠ¯ work on Jacquet integrals and in DiaconisæŠ¯ shuffling theory.

**Branching Coefficients of
Holomorphic Representations**

Zhang Genkai, University of Chalmers, Sweden

We study the restriction to real forms $H/L$ of bounded symmetric domains $G/K$ of the (analytic continuation of) scalar holomorphic discrete series. We find $L$-invariant holomorphic polynomials in terms of the Jack symmetric polynomials and we compute their Segal-Bargmann and spherical transforms.

**Branching Rules and Covariants
of Qubits**

*Roger Howe, Yale University, USA*

The current interest in quantum computation provides a new set of questions in invariant theory. This talk will provide a brief introduction to the ideas of quantum computation, including a desription of the qubit, the basic unit of quantum information. It will then describe some computations in invariant theory which may help elucidate the structure of small systems of qubits.

**Kazhdan-Lusztig Algorithm for
Non-linear Real Reductive Groups and Applications to
Functoriality**

*David Renard, University of Poitiers, France*

Non-linear groups appear in Automorphic Forms theory, for instance via Howe's dual pairs correspondences or Kazhdan-Patterson lifting. Thus, it is natural to try to extend Langlands formalism to these groups, and to study the "functoriality" of the above correspondences. Over the real numbers, we will use the geometric reformulation of Langlands local conjectures due to Adams-Barbasch-Vogan to achieve this goal for double covers of reductive linear groups (e.g. the metaplectic group). This consists of three steps:

- Establish Kazhdan-Lusztig algorithm.
- Establish Vogan's character multiplicity duality.
- Give applications to functoriality on various examples.

Although there is no dual group or $L$-group in the picture, the existence of Vogan's duality for non-linear groups is sufficient for our purposes. It allows us to define $L$-packets and to state functoriality principles.

This is joint work with P. Trapa.

**Holomorphic Continuation of
Generalized Jacquet Integrals**

*Nolan Wallach, University of California at San Diego,
USA*

In this lecture some generalizations of the speaker's earlier work on holomorphic continuation of Jacquet integrals that are general enough to apply to the work that he did on quaternionic representations. In this generalization one must replace standard Bruhat theory with an extension due to Kolk and Varadarajan.

**Measures of entanglement in
quantum computing**

*Nolan Wallach, University of California at San Diego,
USA*

This lecture will give a rapid introduction to quantum computing for mathematicians. The emphasis will be on the notion of entanglement and measures of entanglement. Some new applications of invariant theory will be discussed.

**Classification of Some Classes
of Irreducible Representations of Classical p-adic
Groups**

*Marko Tadic, University of Zagreb, Croatia*

The set of tutorials provide an introduction to classification of some important series of irreducible representations of general linear and classical groups (having in mind unitary representations). We shall deal more with p-adic groups, but we shall also discuss real groups (some of the results are uniform). One of the goals will be to give an introduction to classification modulo cuspidal data of irreducible square integrable representations of classical p-adic groups. Further, we shall talk about unitary duals of general linear groups (and corresponding proofs). We will finish with some questions regarding unitary representations of classical groups.

The topics which we plan to cover are following:

- Harmonic analysis and unitary duals
- Non-discrete locally compact fields, classical groups, reductive groups
- K-finite vectors
- Smooth representations
- Parabolically induced representations
- Jacquet modules
- Filtrations of Jacquet modules
- Square integrable and tempered representations
- Langlands classification
- Geometric lemma and algebraic structures
- Square integrable representations of general linear groups
- Two simple examples of square integrable representations of classical groups
- Invariants of square integrable representations of classical groups
- Reduction to cuspidal lines
- Parameters of representations supported in cuspidal lines
- Integral case
- Non-integral case
- On local Langlands correspondences
- Unitary duals of general linear groups
- On unitarizability problem for classical groups

**Equivariant Analogues of
Zuckerman and Bernstein Functors**

*Pavle Pandzic, University of Zagreb, Croatia*

There are two well known constructions of Harish-Chandra modules: Zuckerman's derived functor construction, and Beilinson-Bernstein localization theory. They both use some homological algebra, however in different categories which makes the relations between them non-obvious. Both constructions can however be performed in the framework of equivariant derived categories introduced by Beilinson-Ginzburg.

In the first talk I will review the notion of (ordinary) derived category, and define the equivariant derived category of Harish-Chandra modules. In the second talk I will explain how to get analogs of Zuckerman and Bernstein functors in the equivariant derived category setting. These functors are related by a version of "Hard Duality" of Knapp and Vogan, which I will sketch at the end.

**Dirac Operators, Lie Algebra
Cohomology and Group Representations**

*Jing Song Huang, Hong Kong University of Science and
Technology, Hong Kong*

The aim of this talk is to show the connection between Dirac cohomology and Lie algebra cohomology of unitary representations.

**The Classical Voronoi
Summation Formula**

*Wilfried Schmid, Harvard University, USA*

Abstract: The Voronoi summation formula provides explicit formulas for sums of the type $\sum_n f(n) a_n$, where $a_n$ is an arithmetically defined sequence of coefficients and $f(x)$ a compactly supported function of bounded variation. It has become an essential tool in analytic number theory. Originally established on a case-by-case basis,it is now regarded as a statement about L-functions satisfying a functional equation, and is deduced from the functional equation. I shall discuss the formula, its proof and some applications.

**A Local Property of
Distributions, with Applications to L-functions**

*Wilfried Schmid, Harvard University, USA*

I shall describe a local property of distributions -- "vanishing to infinite order" at a point -- which can be used to establish functional equations in various contexts. Automorphic distributions for SL(2,R), i.e., boundary distributions of modular forms and Maass forms, have this property, as do certain one-variable distributions derived from automorphic distributions on higher rank groups. This is joint work with Steve Miller.

**L-functions and Voronoi
summation for GL(3)**

*Wilfried Schmid, Harvard University, USA*

Certain L-functions attached to automorphic forms for GL(3,Z), which are important from the point of view of analytic number theory, do not satisfy a functional equation. The classical approach to Voronoi summation does not work in this case. I shall state a Voronoi summation formula for these L-functions, sketch its proof, and describe an application. The same type of arguments can be used as a new method of proof for functional equations and converse theorems for GL(3) and other higher rank groups. This is joint work with Steve Miller.

**K-types of P-eigendistributions**

*Chen-Bo Zhu, National University of Singapore, Singapore*

We study representations of a classical group G which admit certain P-eigendistributions, where P is a parabolic subgroup of G. Through examples, we shall explain how to understand the K-types of G-representations generated by these P-eigendistributions.

**Conformal Geometry and
Global Solutions to the Yamabe Equations on Classical
Pseudo-Riemannian Manifolds**

*Toshiyuki Kobayashi, RIMS, Kyoto, Japan*

The Yamabe operator on a pseudo-Riemannian manifold is a "modified" Laplace-Beltrami operator with scalar curvature involved. The conformal group stabilizes the space of global solutions to the Yamabe equation. Applying this to the flat pseudo-Riemannian manifold R^{p,q}, I shall discuss analytic aspects of minimal representations of indefinite orthogonal groups. This is a joint work with B. Orsted.

**Estimates of Automorphic
forms and Representation Theory
**

*Joseph Bernstein Tel Aviv University, Israel*

**Regularity Theorems for
Automorphic Functionals**

*Joseph Bernstein Tel Aviv University, Israel*

**Analytic Structures on
Representation Spaces of Reductive Groups**

*Joseph Bernstein Tel Aviv University, Israel*

**Restriction of Unitary
Representations**

*Toshiyuki Kobayashi, RIMS, Kyoto, Japan*

In this talk, I will discuss the restriction of unitary representation of a real reductive Lie group G with respect to its reductive subgroup H.

I will focus on the distinguished case where the branching laws do not contain any continuous spectrum, with some motivation, criterion, applications, and examples.

**Finite Dimensional
Representations of Invariant Differential Operators**

*Gerald Schwarz Brandeis University, USA*

Let *G* be a reductive complex algebraic group and * V* a finite dimensional *G*-module. Set *B*:=*D(V) ^{G}*,
the algebra of

*G*-invariant polynomial differential operators on

*V*. One can ask:

1) What is the representation theory of *B*? What
are the primitive ideals of *B*?

2) Does *B* have finite dimensional
representations? In particular, is *B* an
FCR-algebra?

Little is known about these questions when *G* is
noncommutative. We give answers for the adjoint
representation of *SL _{3}(C)*, already an
interesting and difficult case.

**Orbits and Invariants
Associated with a Pair of Commuting Involutions**

*Gerald Schwarz Brandeis University, USA*

**On Zero-free Regions of
L-functions**

*Stephen Gelbart, Weizmann Institute of Science, Israel*

There are (at least) three different approaches towards reaching zero-free regions of the Riemann Zeta-function. One is due to Hadamard and de la Vallee Pousin in the late 1890's, another to Ingham (in 1932) and Balasubramanian - Ramachandra (1976), and the third to Selberg etc. (starting about 1950).

We shall survey the proofs of each of these, and see which can be used to generalize from the zeta-function to arbitrary L-functions defined in terms of automorphic cuspidal representations.

**Lifting of Covariants for
Nilpotent Orbits**

*Kyo Nishiyama, Kyoto University, Japan*

It is known that the lifting of nilpotent orbits for a
reductive dual pair is very useful to study
representations of *G* and/or *G'*. In
this talk, we explain it briefly, and extend it to the
lifting of some equivariant coherent sheaves, which are
regarded to be modules of covariants. Under a reasonable
condition, the lifting will preserve multiplicities,
hence we can get a degree formula of the lifted sheaves
modulo the geometric degree of nilpotent orbits. (Joint
work with Chen-bo Zhu)

**Correspondence of Associated
Varieties of Harish-Chandra Modules**

*Kyo Nishiyama, Kyoto University, Japan*

Let *(G, G')* be a reductive dual pair in the
stable range. We will give a description of good
filtration for the Harish-Chandra modules of *G* and *G'* via the oscillator representation. As a consequence, we can
control the behavior of associated varieties under theta
correspondence. (Joint work with Chen-bo Zhu)

**Multiplicity Free Spaces and
Schur-Weyl-Howe Duality**

*Roe Goodman, Rutgers University, USA*

The following topics will be covered in this set of tutorials:

- Decompositions and Duality for Representations of Reductive Groups
- Commutant Character Formula
- Weyl-Schur Duality and Frobenius Character Formula
- Tensor and Polynomial Invariants for Classical Groups
- Weyl Algebra and Howe Duality
- Spherical Harmonics and Duality
- Brauer Algebra and Harmonic Decomposition of Tensor Spaces
- Seesaw Duality

**The Combinatorics of Multiplicity
Free Spaces**

*Friedrich Knop, Rutgers University, USA*

We define a certain combinatorial structure for each multiplicity free space. Then we show how that determines the spectrum of invariant differential operators via Capelli polynomials. Finally, we explain how to obtain the Capelli polynomials as eigenfunctions of difference operators.

**Special Functions and
Multiplicity Free Spaces**

*Friedrich Knop, Rutgers University, USA*

Every multiplicity free space gives rise to an algebra of difference operators. This algebra contains a copy of sl_2 whose adjoint action integrates to an SL_2-action. We show how this permits the definition of various multivariate generalizations of special functions, in particular Laguerre, Meixner, and Bessel functions.

**Multiplicity-Free Actions and
Tensor Algebras**

*Eng-Chye Tan, National University of Singapore,
Singapore*

We shall use certain realizations of multiplicity-free
actions to study tensor product algebras. Let G be a
reductive group acting in a multiplicity free way on two
algebras ** A** and

**. We can form a tensor product algebra by forming the algebra of covariants of the tensor of these two algebras. It provides information on the decomposition of tensor products of representations coming from**

*B***and**

*A***. We shall illustrate with G as the general linear group.**

*B*

**G _{2}(q) Invariants in
Representations of D_{4}(q)**

*Hung-Yean Loke, National University of Singapore, Singapore*

**Multiplicity Free Actions**

*Chal Benson and Gail Ratcliff, East Carolina University,
USA*

The action of a complex reductive group G on an algebraic variety X is said to be multiplicity free when no irreducible occurs more than once in the associated representation of G on the ring of regular functions on X. We will survey examples and applications for such actions, which provide an organizational principle in Representation Theory. Linear multiplicity free actions will be the main focus in these lectures. Topics will include - examples of multiplicity free decompositions, - criteria for linear multiplicity free actions, - classification of linear multiplicity free actions, - invariant polynomials and polynomial coefficient differential operators, - generalized binomial coefficients, and - spectra for invariant polynomial coefficient differential operators.

**Invariant systems of
differential operators on a Hermitian symmetric space**

*Chal Benson, East Carolina University, USA*

Let G/K be a Hermitian symmetric space. Left G-invariant systems of differential operators on G/K arise from Ad(K)-invariant subspaces of the complexified enveloping algebra for G. This talk will report on recent joint work with D. Buraczewski, E. Damek and G. Ratcliff concerning systems of type (1,1) and their zeros.

**Multiplicity free actions and
analysis on the Heisenberg group**

*Gail Ratcliff, East Carolina University, USA*

Multiplicity free actions occur in the study of Gelfand pairs associated with compact extensions of the Heisenberg group. Results from invariant theory give one information on the spherical functions for such pairs, and the topology of the Gelfand space.

**Vertex operators and McKay
correspondence**

*Nai-Huan Jing North Carolina State University*

In early 1980's McKay discovered that simply laced Dynkin diagrams can be canonically and conceptually obtained from Kleinian subgroups, the finite subgroups of SL(2, C). The correspondence can be traced back to Platonian solids dated 2000 years ago. McKay correspondence has played a role in algebraic geometry (Nakajima's revolutionary work) and Lie groups (eg. Steinberg, Kostant and Vogan's works). We will give a new approach to the correspondence based on affine Lie algebras and their representations, in which the affine Dynkin diagrams are manifested in the study. We show that the representation theory of wreath products of a Kleinian subgroup and the symmetric groups give naturally all the structures via vertex operators. More interestingly our approach can be easily generalized to quantum affine algebras as well as some distinguished double covering groups of the generalized symmetric groups based on ideas from affine Lie algebras and vertex operator representations. The talk is based on my work in 1989 and recent joint work with I. Frenkel and W. Wang.

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