Avraham (Rami) Aizenbud 
אברהם (רמי) איזנבוד 










I am working on representation theory and related questions of geometry and analysis. Most of my research until now can be divided into 2 parts:
In my work I use various tools from representation theory, algebraic geometry, analysis and algebra. My results have applications in diﬀerent areas of representation theory and the theory of automorphic forms. ContentsContents
1. Distributions 1.1. The wave front set 1.2. Uncertainty Principles 1.3. Localization Principle 1.4. Vanishing of equivariant distributions 1.5. Comparison of orbital integrals 1.6. Homology of spaces of Schwartz sections 2. Representation theory of reductive groups over local ﬁelds 2.1. Harmonic analysis on spherical spaces 2.2. Derivatives of representations of GL_{n} 3. Other topics of my research 3.1. A quantum analog of Kostant theorem 3.2. Morse theory References 1. DistributionsThe theory of distributions and, in particular, the study of invariant distributions are important tools in representation theory. We are interested in the study of distributions both as a topic as itself and as a useful tool in representation theory. One can divide the study of distributions into 2 cases:
In both cases the space of distributions is described as the space of functionals on a certain space of test functions on certain geometric objects. In the Archimedean case the underlying geometric objects are smooth manifolds. Often one would like to restrict the study to a narrower generality such as real algebraic manifolds or, more generally, Nash (i.e. smooth semialgebraic) manifolds (see [Shi87] for description of the theory of Nash manifolds). The reason that one wants to consider this generality is that it allows to give some growth conditions on the distributions. Distributions satisfying those growth conditions are called Schwartz distributions, and the corresponding test functions are called Schwartz functions, see for more details [AG08]. In the nonArchimedean case the underlying geometric objects are lspaces, i.e. locally compact totally disconnected Hausdorﬀ spaces, and the test functions are locally constant compactly supported functions. They are also referred to as Schwartz functions. For the the description of the theory of Schwartz distributions over lspace see [BZ76]. Note that in this case there is no distinction between Schwartz distributions and general distributions. Often one would like to restrict study of distributions to a narrower generality such as analytic nonArchimedean manifolds (in the sense of Serre). This allows one to use tools that rely upon the existence of local model. In both cases the space of Schwartz functions is denoted by S(X) and the space of Schwartz distributions is denoted by S^{∗}(X), where X is the underlying geometric object. The notion of singular support (a.k.a. the characteristic variety) plays an important role in the theory of invariant distributions over real manifolds. The notion of singular support appears in the theory of Dmodules. The singular support of a distribution ξ on a real algebraic manifold X is deﬁned to be the singular support of the Dmodule it generates. The singular support of ξ is a closed subset of the cotangent bundle T^{∗}X. Roughly speaking, the singular support of ξ over each point x ∈ X is a cone in the cotangent space T_{x}^{∗}(X) that measures in what directions the distribution ξ has singularities at x, in terms of the diﬀerential equations ξ satisﬁes. In the nonArchimedean case the theory of Dmodules is not available, since diﬀerential operators do not act on distributions. Therefore, the notion of singular support is also not directly available. However, in both cases one has a similar notion: the notion of the wave front set. The wave front set of a distribution ξ is also a closed subset of the cotangent bundle T^{∗}X. Similarly to the the above, wave front set measures singularities of a distribution, but using local Fourier transform instead of Dmodules. In the Archimedean case, the singular support and the wave front set are closely related. In particular, the wave front set is a subset of the singular support. While the wave front is available in both cases, its study is much more diﬃcult than the study of the singular support since we do not have the theory of Dmodules for it. Usually, the theory of distributions in the nonArchimedean case is simpler than in the Archimedean case. However, the lack of the theory of Dmodules reverses the situation in some aspects. We am interested in using wave front set in nonArchimedean setting to transfer results on distributions proved in the Archimedean case with the help of singular support. 1.1.1. The Integrability Theorem. An important theorem in the theory of Dmodules is the Integrability Theorem. This theorem states that the singular support of a Dmodule is coisotropic (a.k.a integrable). This theorem was proved in [GQS71, KKS73, Mal79, Gab81]. The integrability roughly means that the singular support is large (provided it is not empty). In particular, the Integrability Theorem implies the Bernstein inequality that states that the dimension of the singular support is at least the dimension of the underlying space. The Integrability Theorem has a direct application in the theory of distributions in the Archimedean case. Namely, it implies that the singular support of a distribution is coisotropic. However, this approach can not be used in the nonArchimedean case where one has to consider the wave front set instead. In [Aiz] we have studied the wave front set in the nonArchimedean case. We introduced the notion of “weakly coisotropic” subvariety of T^{∗}X and proved the following partial analog of the Integrability Theorem. Theorem I. Let ξ be a distribution on analytic manifold X over nonarchimedean ﬁeld. Then the wave front set of ξ is weakly coisotropic subvariety of T^{∗}X. This result is signiﬁcantly weaker than the conjectural full analog of the Integrability Theorem. Its proof is also much simpler than the proof of the Integrability Theorem. However, for many applications such as those described in §§1.2, §§1.4, it can replace the Integrability Theorem. The reason is that nonempty weakly coisotropic variety contains a nonempty coisotropic variety (and therefore also a Lagrangian subvariety). 1.1.2. Wave front set of a Fourier transform of algebraic measures. Using the theory of Dmodules, Bernstein has proved the following theorem: Theorem 1.1.1 ([Ber72]). Let p be a polynomial on a real vector space V . Consider p as distribution on V and consider its Fourier transform F(p) as a distribution on V ^{∗}. Then F(p) is a smooth function on an open dense set U ⊂ V ^{∗}. Again, this approach cannot be directly applied in the nonArchimedean case. In [AD] we proved an analog of this theorem. Namely, we introduced the notion of WFholonomic distribution, i.e. a distribution whose wave front set is small in an appropriate sense. Then we prove the following theorem. Theorem II. Let p be a polynomial on a vector space V over nonArchimedean ﬁeld. Consider p as distribution on V and consider its Fourier transform F(p) as a distribution on V ^{∗}. Then F(p) is WFholonomic. This implies that it is a smooth function on an open dense set U ⊂ V ^{∗}. Remark.
An uncertainty principle is a claim that certain restrictions on a distribution on a vector space are inconsistent with certain restrictions on its Fourier transform. The most wellknown example of such principle is the Heisenberg uncertainty principal. This principle is quantitative. We are mostly interested in qualitative uncertainty principles. One can consider the Integrability Theorem as an example of such a principle. Namely, restrictions on the support of a distribution together with restrictions on the support of its Fourier transform gives restrictions on its singular support (resp. wave front set). These restrictions are often inconsistent with integrability. In our work we use diﬀerent kinds of uncertainty principles. Occasionally (see [AGS08, AG09a, Aiz, AGb]) we developed these principles further. In many cases one can localize the study of equivariant distributions. In the nonArchimedean case very general localization principle was proved in [Ber82]. Namely, given a map p : X → Y , one can reduce the study of equivarint distributions on X to the study of equivariant distributions on each of the ﬁbers p^{−1}(y) for each y ∈ Y . A complete analog of this result in the Archimedean case seems to be far from reach. However, in [AG09a, AGb, AOSa, AOSb] we proved several partial analogs of this theorem. 1.4. Vanishing of equivariant distributions. Many questions of representation theory can be reduced to proving that certain space of equivariant distributions vanishes. More precisely, we study a group G acting on a manifold X and a character χ of the group G, and we interested in showing that Most of the works [AGRS10, AGS08, AG09a, AG10b, AG09b, AGa, AS12, AOSa, AOSb] and a part of the work [Aiz] are devoted to diﬀerent instances of this problem. In most of these works we have followed the strategy described in [AG09a]. Namely, we linearize the space X. Then we prove, using an inductive argument, that any distribution in S^{∗}(X)^{G,χ} has to be supported in a small closed subset of X that we call the singular set. Then we use some more complicated geometric tools (such as the localization principle) along with nongeometric tools based on Fourier transform (such as various kinds of uncertainty principles) in order to conclude vanishing of S^{∗}(X)^{G,χ}. This strategy ﬁts best for reductive groups. Unipotent groups require diﬀerent methods as developed in [AGb, AOSa, AOSb]. In the works listed above we analyze many diﬀerent spaces of equivariant distributions. In particular, we have established the following results
Remark.
1.5. Comparison of orbital integrals. As it was mentioned earlier, one of the aims of my research is the study of the space S^{∗}(X)^{G} of Ginvariant distributions on X. One can consider a dual question — the study of the space S(X)_{G} of the G coinvariants of the representation S(X). In the nonArchimedean case these two problems are equivalent since S^{∗}(X)^{G} = (S(X)_{G})^{∗}. However, in the Archimedean case the second space contains slightly more information since S(X)_{G} does not have to be Hausdorﬀ. One way to study the space S(X)_{G} is via regular orbital integrals. Namely, in many cases we have a categorical quotient X → X∕∕G. Let (X∕∕G)_{reg} be the set of regular values of this map and X_{reg} be its preimage. Usually the set (X∕∕G)_{reg} consists of closed orbits of maximal dimension. The set X_{reg} is the union of these orbits. Let Ω_{X,G} : S(X) → C^{∞}((X∕∕G)_{reg}) be the direct image (more precisely, it is the restriction to X_{reg} followed by the direct image). Clearly Ω_{X,G} factors through S(X)_{G}. In many cases (more precisely, when we have “density of regular orbital integrals”) Ω_{X,G} gives an isomorphism between S(X)_{G} and the image Ω_{X,G}(S(X)). One can often make a similar construction for the twisted case (when we are given a character ψ of G and study the coequvivariants S(X)_{G,ψ}). It is usually very hard to explicitly describe the space Ω_{X,G}(S(X)). However, it often happens that Ω_{X,G}(S(X)) is the same for diﬀerent pairs X and G. Namely, suppose we have another group G′ acting on a space X′ so that the categorical quotients X∕∕G and X′∕∕G′ are equal. Then one can ask whether the spaces Ω_{X,G}(S(X)) and Ω_{X′,G′}(S(X′)) coincide. More generally, one can ask whether there exists a matching factor γ ∈ C^{∞}((X∕∕G)_{reg} so that Ω_{X,G}(S(X)) = γ ⋅ Ω_{X′,G′}(S(X′)). This phenomenon is very useful for the Langlands program, more speciﬁcally, for comparison of representations of diﬀerent groups. Let G be a reductive group. The characters of representations of G are Ad(G)invariant distributions on G. HarishChandra proved that these characters span (in an appropriate sense) the space of invariant distributions S^{∗}(G)^{Ad(G)}. Often one can relate the geometric quotient G∕∕Ad(G) with G′∕∕Ad(G′) where G′ is another group. In this case on can use the study of orbital integrals in order to compare S^{∗}(G)^{Ad(G)} and S^{∗}(G′)^{Ad(G′)}. This gives a relation between the (complexiﬁed and completed) Grothendieck group of representations of G and the (complexiﬁed and completed) Grothendieck group of representations of G′. Often one can use the trace formula in order to deduce from this a relation between the sets of irreducible representations of G and of G′. One can use similar considerations when studying “relative representation theory” (see §§2.1). In this case one will use the relative trace formula and will compare the categorical quotients H_{2}∖∖G∕∕H_{1} with H′_{2}∖∖G′∕∕H′_{1}. In the work [AGb], we establish the following equality of spaces of orbital integrals Theorem IV. Let N_{n} be the group of n × n upperunipotent matrices. Let N(ℝ) × N(ℝ) act on GL_{n}(ℝ) by (n_{1},n_{2})(x) = n_{1}^{t}xn_{2}. Let A ⊂ GL_{n} be the set of diagonal matrices. Let _{1} : S(GL_{n}(ℝ)) → C^{∞}(A) be the map deﬁned by Then Im(_{2}) = Im(_{1}) The nonArchimedean counterpart of this theorem was done in [Jac03]. An application of these works can be found in [FLO]. 1.6. Homology of spaces of Schwartz sections. Study of coinvariants S(X)_{G} can be generalized in two diﬀerent directions.
We believe that the study of H_{∗}(G,S(X,E)) is a fundamental mathematical question and an essential part of my work is directly related to this question. As complicated as this problem may be in the general case, it is rather simple in the case when X is a single orbit. In this case the homology can be computed explicitly. This computation is often called the Shapiro lemma. It was clasically done in a diﬀerent setting. In the nonArchimedean case it follows directly from the Grothendieck lemma (see e.g. [AAG]). In the Archimedean case this is more problematic since the category of Grepresentations is not abelian. In case that G is contractible, one can replace it by its Lie algebra. In this case we proved it in [AG10a] and some generalisation of it in [AGS]. 2. Representation theory of reductive groups over local ﬁeldsSimilarly to the theory of distributions, representation theory of reductive groups over local ﬁelds is also divided into the Archimedean and nonArchimedean case. In what follows we ﬁx a local ﬁeld F (i.e. a locally compact nondiscrete topological ﬁeld). The ﬁeld F might be Archimedean (i.e. ℝ or ℂ) or nonArchimedean. Also, for an algebraic variety X, we will consider X(F) as a topological space. In the nonArchimedean case, it will be an lspace and in the Archimedean case (for smooth X), it will be a real manifold. In case there is no possible confusion, we will denote X(F) simply by X. Let G denote an algebraic group deﬁned over F (unless stated otherwise G will be reductive). By “representation of G” we will mean a representation of G(F). We are interested in the study of smooth Grepresentations. In the nonArchimedean case these are representations in a complex (usually ∞dimensional) vector space such that each vector has an open stabilizer. In the Archimedean case this means a topological representation in a Frechet space V which is continuous w.r.t. each of the seminorms and such that for any vector v ∈ V the action map G → V is diﬀerentiable. In the Archimedean case one also has to impose an additional condition — admissibility. In the Archimedean case there is another category of representations — the category of HarishChandra modules. By CasselmannWalach theorem (see [Wal92], §§§11.6.8) the category of HarishChandra modules is equivalent to the category of smooth admissible representations. However, certain constructions give diﬀerent answers in those categories. We will be manly interested in the category of smooth admissible representations and we will view the category of HarishChandra modules as a tool for its study. 2.1. Harmonic analysis on spherical spaces. One can view representation theory of a given group G as harmonic analysis on G, viewed as a space, with respect to the twosided action of the group G × G. For example, if G(F) is a compact group, the PeterWeyl Theorem states that the decomposition into direct sum of the space L^{2}(G) with respect to the action of G × G is labeled by the irreducible representations of G. However, for noncompact G this point of view is limited, since not every type of representation theory of G can be studied in terms of harmonic analysis of a certain space of functions on G. In particular, it seems impossible to study in such a way general unitary representations. Nevertheless, many questions of representation theory can be reformulated in the language of harmonic analysis on G. Let H ⊂ G be a subgroup. Following the above point of view, we may consider harmonic analysis on G∕H with respect to the action of G as a generalization of representation theory. In order to obtain the representation theory of a group G′, one should substitute G := G′× G′ and H := ΔG′ – the diagonal copy of G. In what follows, we will refer to harmonic analysis on G∕H as “relative representation theory” or “representation theory of the pair (G,H)”. It seems unpractical to consider harmonic analysis on G∕H as a generalization of representation theory for arbitrary H, therefore we will always put some assumptions on H. One popular assumption is that H is a symmetric subgroup, i.e. there exists an involution on G such that H is the set of its ﬁxed points. A weaker assumption is that H is a spherical subgroup, i.e. there exists a Borel subgroup B of G such that HB is open. In what follows, we will consider only those cases and we will refer to the pair (G,H) and the space G∕H as symmetric or spherical. We have studied several problems in harmonic analysis on spherical spaces G∕H. Most of them are translations of known results of representation theory to the case of a pair. These problems are interesting not only as natural analogs of fundamental results, but they also have applications in classical questions of representation theory, such as classiﬁcation of representations and constructions of models of representations, and in topics of the theory of automorphic forms such as study of automorphic periods and the relative trace formula. 2.1.1. Gelfand Pairs. The ﬁrst fundamental result in representation theory is Schur’s lemma. When we translate this result to the relative case we obtain the following property: Deﬁnition 2.1.1. A pair (G,H) consisting of a group and a subgroup is said to be a Gelfand pair if any irreducible representation π of G is “included” in S(G(F)∕H(F)) with multiplicity at most 1. Formally speaking, it means that dimHom(S(G(F)∕H(F)),π) ≤ 1 or, equivalently, dimHom_{H(F)}(π_{H(F)}, ℂ) ≤ 1. If we replace the trivial representation with a character ψ, the pair is called twisted Gelfand pair w.r.t. ψ. Note that if a pair (G,H) is a Gelfand pair, then the “decomposition” of S(G(F)∕H(F)) to irreducible representations is unique. Gelfand pairs have various applications to classical questions of representation theory and harmonic analysis. These include the classiﬁcation of representations and constructing canonical bases for irreducible representations and spaces of functions on homogeneous spaces. More recent applications of Gelfand pairs are in the theory of automorphic forms, for instance in the splitting of automorphic periods and in the relative trace formula. Some of these applications are described in [Gro91]. There is a stronger version of the notion of a Gelfand pair: Deﬁnition 2.1.2. A pair (G,H) is said to be a strong Gelfand pair if any irreducible representation π of G(F), when restricted to H(F), “includes” any irreducible representation of H with multiplicity at most 1. Formally speaking, this means that for any irreducible representation ρ of H, we have Hom_{H(F)}(π_{H(F)},ρ) ≤ 1. The notion of Gelfand pair and strong Gelfand pair are connected in the following way: Proposition 2.1.3. A pair (G,H) is a strong Gelfand pair if and only if the pair (G × H,ΔH) is a Gelfand pair. The main tool to prove that a pair (G,H) is a Gelfand pair is the following criterion by Gelfand and Kazhdan ([GK75]). Theorem 2.1.4. Suppose that there exists an involutive antiautomorphism σ of G such that any distribution on G which is invariant with respect to the H × H twosided action is invariant with respect to σ. Then the pair (G,H) is a Gelfand pair. This criterion implies an analogous criterion for strong Gelfand pairs. Using the results discribed in §1.4 and the GelfandKazhdan Criterion, we have proved the following theorem
Apart of the GelfandKazhdan method, there are other methods for proving the Gelfand property. One such method, that will be discussed in section 2.1.3, is to transfer results about the Gelfand property from zero characteristic to positive characteristic. This is useful, since some of the tools that we have used for proving the condition of the GelfandKazhdan criterion are based on the Jordan decomposition and the Luna slice theorem which are problematic in positive characteristic. Another useful method for proving the Gelfand property is averaging of functionals. An important construction in representation theory is averaging with respect to the action of a subgroup H_{1} ⊂ G, of an H_{2}invariant functional on a representation of G, where H_{2} ⊂ G is another subgroup. Many constructions in representation theory and automorphic forms (such as intertwining operators, periods of automorphic forms, certain Lfactors, etc.) can be viewed as a special case of the above construction. Note that often this integral does not converge and one has to regularize it, usually using analytic continuation. One can use this construction in order to deduce the Gelfand property of one pair from the Gelfand property of another pair. We implemented this method in the work [AGJ09] where we deduced the uniqueness of Shalika functionals in the Archimedean case from the Gelfand property of the pair (GL_{n+k},GL_{n} ×GL_{k}). Namely, we have proved the following theorem. Theorem VI. Let P_{n,n} ⊂ GL_{2n} be the maximal parabolic subgroup corresponding to the partition (n,n) and let p : P_{n,n} → M_{n,n} = GL_{n} × GL_{n} be the projection to its Levi factor. Let GL_{n} ⊂ M_{n,n} be the diagonal and let S := p^{−1}(GL_{n}). Let F be an Archimedean ﬁeld. Let ψ be a generic additive character of S(F). Then the pair (GL_{2n}(F),S(F)) is twisted Gelfand pair w.r.t. ψ. The nonArchimedean counterpart of this theorem was done in [JR96]. In addition, We believe that the methods that will be discussed in section 2.1.2 will be very helpful in proving the Gelfand property of spherical pairs. 2.1.2. CohenMacaulay property. There are several phenomena in representation theory and harmonic analysis which are hard to formalize exactly and on the ﬁrst glance do not necessarily look related. We believe that these phenomena are related to a CohenMacaulay property. In this section we will discuss only the nonArchimedean case. Let us shortly describe some of those phenomena:
Let us now recall the CohenMacaulay property: Deﬁnition 2.1.5. A ﬁnitely generated module M over a commutative algebra A (ﬁnitely generated over a ﬁeld) is called CohenMacaulay of dimension d if there exists a polynomial subalgebra B ⊂ A of rank d such that M is free and ﬁnitely generated over B. The following theorem is well known (see e.g. [BBG97] section 2). Theorem 2.1.6. A ﬁnitely generated module M over a commutative algebra A is CohenMacaulay of dimension d if and only if for any smooth and ddimensional algebra B acting on M, commuting with the action of A, the module M is projective over B. This theorem implies that the notion of CohenMacaulay module is stable when one changes the algebra. Namely, we have the following corollary. Corollary 2.1.7. Let M be a module over a commutative algebras A and B. s.t. the actions of these algebras commute and M is ﬁnitely generated over both. Than M is CohenMacaulay over A if and only if M is CohenMacaulay over B. This ﬂexibility allows one to deﬁne the notion of CohenMacaulay module in diﬀerent abelian categories such as the category of Grepresentations. We believe that the phenomena described above are all related to the following conjecture: Conjecture 1. The module S(G(F)∕H(F)) is a (locally) CohenMacaulay object in the category of smooth Grepresentations M(G). In the group situation this conjecture specializes to the following theorem. This theorem follows from Bernstein’s second adjointness theorem (see [Ber]). One can reformulate this theorem in the following way: Theorem 2.1.9 (Bernstein). For any congruence subgroup K_{n} ⊂ G(F), the Hecke algebra H_{Kn}(G), of double K_{n} invariant measures on G(F), is a CohenMacaulay module over its center. In many cases (for example, the case G = GL_{n}) the Bernstein center is regular. In these cases the above theorem implies that H_{Kn}(G) is locally free over its center. This theorem has also analog in the Archimedean case, see [BBG97]. In [AS] we proved Conjecture 1 for certain special cases and relate it to the phenomena above. Namely, we have proved the following theorem
2.1.3. Spherical Pairs over Close Local Fields. There are several methods that allow one to relate problems over ﬁelds of positive characteristic with problems over ﬁelds of zero characteristic. Most of these methods are based on approximating a ﬁeld of zero characteristic with ﬁelds of positive characteristic. There is a diﬀerent method developed in [Kaz86], which is based on approximating a local ﬁeld of positive characteristic with local ﬁelds of zero characteristic. In this work the following theorem is proved: Theorem 2.1.10 (Kazhdan). Let G be a reductive group that splits over ℤ. Let K_{n} be a congruence subgroup. Then the Hecke algebra H_{Kn}(G) does not change when we replace F with a “close enough” local ﬁeld F′. This theorem means that the representation theory of G(F), when F is a ﬁeld of positive characteristic, can be approximated by the representation theory of G(F′), where F′ is a ﬁeld of zero characteristic. In the work [AAG], we prove the following analog of this theorem. Theorem VIII. Let (G,H) be a spherical pair which satisﬁes certain assumptions. Then the Hecke module S(G(F)∕H(F))^{Kn} does not change when we replace F with a “close enough” local ﬁeld F′. This theorem allows us to deduce the following corollary from Theorem VII:
In the proof of Theorem VIII we used the fact that the Hecke module is ﬁnitely generated over the Hecke algebra. We also proved the following criterion for this fact: Theorem X. Let (G,H) be a spherical pair. Then the following are equivalent:
The proof in [AAG] relies on the theory of the Bernstein center (see [BD84]) and on certain smoothness analysis of some group schemes over local rings. 2.1.4. Distinguished representations with respect to a symmetric subgroup. Another fundamental question in representation theory of a pair (G,H) is characterization of Hdistinguished representations of G, i.e. those representations π satisfying Hom_{H}(π_{H}, ℂ)≠0. In general, this question is very complicated and in some cases carries an arithmetical content (for one such case, see [GP94] and [Wala]). However, for symmetric pairs the following partial answer to this question is expected: Conjecture 2. Let (G,H) be a symmetric pair, and let τ be an involutive automorphism of G such that H = G^{τ}. Let π be an irreducible Hdistinguished representation of G. Then the Lpacket of π is invariant with respect to the functor π∘τ, where denotes the smooth dual representation. Theorem 2.1.11. Let G be a reductive group and let τ be its involutive automorphism. Set H := G^{τ}. Assume (G,H) satisﬁes the GelfandKazhdan criterion. Let π be an Hdistinguished irreducible representation of G. Then π ∘ τ. A special case of this theorem was proved in [JR96], as part of Theorem 1.1, and one can easily prove this theorem along the same lines. More challenging is the case when the symmetric pair does not satisfy the GelfandKazhdan criterion. In [AL] we have studied one such case. We have proved the following theorem. Theorem XI. Let G = GL(V ) where V is an hermitian space over ℂ. Let τ be the involution given by τ(x) = (x^{t})^{−1} and let σ be given by σ(x) = x^{t}. In this case H := G^{τ} is the unitary group. Than for any Hdistinguished G representation we have π ∘ τ The nonArchimedean case is done in [FLO]. 2.2. Derivatives of representations of GL_{n}. The theory of derivatives of representation of GL_{n} was developed in [BZ77] in the nonArchimedean case. It became an important tool in representation theory of GL_{n} and particularly in the study of the internal structure of those representations, see e.g. [Zel80, Tad86]. A decade later the theory of derivatives was adapted to the study of unitary representations of GL_{n} in the Archimedean case, see [Sah89, Sah90, SS90]. In [AGS] we generalized this theory to arbitrary smooth representations. The theory of derivatives is based on analysis of representation theory of the mirabolic group P_{n} ⊂ GL_{n}, i.e. the subgroup of matrices with the last row (001). In some sense, one can reduce the study the of representations of P_{n} to the study the of representations of GL_{n−1} and of P_{n−1}. By induction this assigns to any representation π of P_{n} a sequence of representations D^{i}(π) of GL_{n−i}. These representations called the derivatives of π. One deﬁnes derivatives of representations of GL_{n} by restricting them to P_{n} and then taking derivatives. It turns out that the highest (nonzero) derivative plays a special role. A representation is of depth d if its dth derivative is the highest (nonzero) derivative. The above description can be made rigorous in the nonArchimedean case. However, the Archimedean case is aggravated by the following diﬃculties:
If we are only interested in unitary representations, then the ﬁrst diﬃculty can be overcome since there is nice a category of unitary representations of P_{n}. The second diﬃculty also turns out to be irrelevant. However, this approach will only give the higher derivative. This was done in [Sah89], and this highest derivative was called the adduced representation. Due to (1) we were unable to directly use in [AGS] the picture described above, but rather deﬁned the derivative using the action of the Lie algebra. We took into account (2) by replacing the “ﬁber” with the “jet” which is intermediate object between the ﬁber and the stalk (a similar construction was used by Casselmann in his version of Jacquet functor in the Archimedean case). We proved some basic properties of the derivative. In particular, we proved the following theorem. Theorem XII. Let M_{∞}(G_{n}) denote the category of smooth admissible Fréchet representations of moderate growth and let M_{∞}^{d}(G_{n}) denote the subcategory of representations of depth ≤ d. Then
We applied this theorem for the study of adduced representations and the study of degenerate Whittaker functionals which are generalization of the classical Whittaker functionals. More precisely,
3. Other topics of my research3.1. A quantum analog of Kostant theorem. A fundamental result in representation theory of Lie algebras is Kostant’s theorem saying that the algebra of polynomial functions on a reductive Lie algebra is free as a module over its invariants. In [AY11] we prove a quantum analog of this theorem for the general linear group. Namely, we proved the following theorem. Theorem XIII. The quantum deformation O_{q}(Mat_{n}) of the algebra of polynomials on Mat_{n} is free over the subalgebra of invariants under the adjoint coaction of O(GL_{q}(n)), for q not a root of unity or q = 1. There are other quantum versions of Kostant’s theorem, see [JL94, Bau00]. Our result is stronger, however, unlike [JL94, Bau00], it is restricted to the Mat_{n} case. In fact, we can not even formulate our result for other cases since it is not clear what is a natural quantum deformation of the algebra of polynomials on a general reductive algebra. Additionally, I am interested in diﬀerential topology and, in particular, Morse theory. Morse theory assigns a chain complex to a Riemannian manifold with a Morse function. The homology of this complex is the homology of the manifold. In [AZ] we investigated the functorial properties of this assignment. References
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