We compute the Jantzen filtration of a $\mathcal {D}$-module on the flag variety of $\operatorname {\mathrm {SL}}_2(\mathbb {C})$. At each step in the computation, we illustrate the $\mathfrak {sl}_2(\mathbb {C})$-module structure on global sections to give an algebraic picture of this geometric computation. We conclude by showing that the Jantzen filtration on the $\mathcal {D}$-module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson and Bernstein.

We construct an action of the affine Hecke category on the principal block $\mathrm {Rep}_0(G_1T)$ of $G_1T$-modules where G is a connected reductive group over an algebraically closed field of characteristic $p> 0$, T a maximal torus of G and $G_1$ the Frobenius kernel of G. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs.

Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$ . A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$ . This generalises a result of Vogan [Vog91].

We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G.

Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20].

Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finite-dimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category ${\mathcal{O}}$ and Koszul duality.

Let $G$ be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces to the invariant vectors coming from webs. In the case of $G= \mathrm{SL} (3)$, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is $\mathrm{CAT} (0)$, is explained by the fact that affine buildings are $\mathrm{CAT} (0)$.

We establish a character multiplicity duality for a certain natural class of nonlinear (nonalgebraic) groups arising as two-fold covers of simply laced real reductive algebraic groups. This allows us to extend part of the formalism of the local Langlands conjecture to such groups.

For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.

Write ${{\Theta }^{E}}$ for the stable discrete series character associated with an irreducible finite-dimensional representation $E$ of a connected real reductive group $G$. Let $M$ be the centralizer of the split component of a maximal torus $T$, and denote by ${{\Phi }_{M}}\left( \gamma ,\,{{\Theta }^{E}} \right)$ Arthur’s extension of $|D_{M}^{G}\,\left( \gamma \right)|{{\,}^{1/2}}\,{{\Theta }^{E}}\,\left( \gamma \right)$ to $T\left( \mathbb{R} \right)$. In this paper we give a simple explicit expression for ${{\Phi }_{M}}\left( \gamma ,\,{{\Theta }^{E}} \right)$ when $\gamma $ is elliptic in $G$. We do not assume $\gamma $ is regular.

We use the geometry of characteristic cycles of Harish-Chandra modules for a real semi-simple Lie group $G_{\mathbb{R}}$ to prove an upper triangularity relationship between two bases of each special representation of a classical Weyl group. One basis consists of Goldie rank polynomials attached to primitive ideals in the enveloping algebra of the complexified Lie algebra ${\mathfrak{g}}$; the other consists of polynomials that measure the Euler characteristic of the restriction of an equivariant line bundle on the flag variety for ${\mathfrak{g}}$ to an irreducible component of the Springer fiber. While these two bases are defined only using the structure of the complex Lie algebra ${\mathfrak{g}}$, the relationship between them is closely tied to the real group $G_{\mathbb{R}}$. More precisely, the order leading to the upper triangularity result is a suborder of closure order for the orbits of the complexification of a maximal compact subgroup of $G_{\mathbb{R}}$ on the flag variety for ${\mathfrak{g}}$.

For the groups $G={\mathrm{Sp}}(p,q),\ \mathrm{SO}^\ast(2n)$, and $\mathrm{U}(m,n)$, we consider degenerate principal series whose infinitesimal character coincides with a finite-dimensional representation of $G$. We prove that each irreducible constituent of maximal Gelfand–Kirillov dimension is a derived functor module. We also show that at an appropriate ‘most singular’ parameter, each irreducible constituent is weakly unipotent and unitarizable. Conversely we show that any weakly unipotent representation associated to a real form of the corresponding Richardson orbit is unique up to isomorphism and can be embedded into a degenerate principal series at the most singular integral parameter (apart from a handful of very even cases in type D). We also discuss edge-of-wedge-type embeddings of derived functor modules into degenerate principal series.

We obtain a decomposition formula of a representation of Sp(p, q) or SO*(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules. In particular, we show that such an induced representation is decomposed into a direct sum of irreducible unitarily induced modules from derived functor modules under some regularity condition on the parameters. In particular, representations of SO*(2n) and Sp(p, q) induced from one-dimensional unitary representations of their parabolic subgroups are irreducible.

We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group $K={{O}_{p}}\left( C \right)\times {{O}_{q}}\left( C \right)$ acts on the space ${{M}_{p,\,q}}\,\text{of}\,p\,\times \,q$ complex matrices by $\left( a,b \right)\cdot x=ax{{b}^{-1}}$, and so does its identity component ${{K}^{0}}=\text{S}{{\text{O}}_{p}}\left( \text{C} \right)\times \text{S}{{\text{O}}_{\text{q}}}\left( \text{C} \right)$. A $K$-orbit (or ${{K}^{0}}$-orbit) in ${{M}_{p,q}}$ is said to be nilpotent if its closure contains the zero matrix. The closure, $\bar{\mathcal{O}}$, of a nilpotent $K$-orbit (resp. ${{K}^{0}}$-orbit) $\mathcal{O}$ in ${{M}_{p,q}}$ is a union of $\mathcal{O}$ and some nilpotent $K$-orbits (resp. ${{K}^{0}}$-orbits) of smaller dimensions. The description of the closure of nilpotent $K$-orbits has been known for some time, but not so for the nilpotent ${{K}^{0}}$-orbits. A conjecture describing the closure of nilpotent ${{K}^{0}}$-orbits was proposed in $[11]$ and verified when $\min \left( p,\,q \right)\le 7$. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to $\mathcal{O}$ and determination of the basic relative invariants of these spaces.

The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra $\mathfrak{s}\mathfrak{o}\left( p,\,q \right)$ under the adjoint action of the identity component of the real orthogonal group $\text{O}\left( p,\,q \right)$.

In the first part of this paper generalizations of Hesselink’s $q$-analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a $q$-analog of the Kostant-Rallis theorem is given for the real group $\text{SL(4,}\,\mathbb{R}\text{)}$ (that is $\text{SO}(4)$ acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

We give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties X in the classical Hermitian symmetric spaces: namely the intersection homology sheaves IHX and the constant sheaves ℂX. The three main cases of interest are the Hermitian symmetric spaces for groups of type An (the standard Grassmannian), Cn (the Lagrangian Grassmannian) and Dn. In particular we find that CC(IHX) is irreducible for all Schubert varieties X if and only if the associated Dynkin diagramis simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the Cn and Dn cases are new.

Our approach is to compute CC(ℂX) by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute CC(IHX). The geometric method is based on the fundamental formula where the Xr ↓ X constitute a family of tubes around the variety X. This formula leads at once to an expression for the coefficients of CC(ℂX) as the degrees of certain singular maps between spheres.