Vogan and Barbasch-Vogan attach two similar invariants to representations of a reductive Lie group, one by an algebraic process, the other analytic. They conjectured that the two invariants determine each other in a definite manner. Here we prove the conjecture. Our arguments involve two finer invariants -- the characteristic cycles of representations -- which are interesting in their own right.
In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups.
Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.
If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $τ$ is a unitary representation of $H$, then the authors give a sufficient condition on $ξ\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^Gτ$. In the special case where $τ$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $π$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\operatorname{WF}(π)$ in terms of the direct integral decomposition of $π$ int…
With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front se…
We define and study the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles Bun_G for an arbitrary reductive group G; its definition is given in terms of the Vinberg semigroup of G, and is due to Drinfeld (unpublished). Throughout the article we prefer to view the compactification as a canonical multi-parameter degeneration of Bun_G which we call the Drinfeld-Lafforgue-Vinberg degeneration VinBun_G. We construct local models for the degeneration VinBun_G which "factorize in families" and use them to study its singularities, generalizing results of the article [Sch1] wh…
Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation pi such that O occurs in the wave front cycle of pi. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space.
In this paper we first construct natural filtrations on the full theta lifts for any real reductive dual pairs. We will use these filtrations to calculate the associated cycles and therefore the associated varieties of Harish-Chandra modules of the indefinite orthogonal groups which are theta lifts of unitary lowest weight modules of the metaplectic double covers of the real symplectic groups. We will show that some of these representations are special unipotent and satisfy a K-type formula in a conjecture of Vogan.
Flag domains are open orbits of real forms $G_\mathbb{R}$ of complex reductive Lie supergroups $G$ in $G$-flag supermanifolds $Z = G/P$. This thesis discusses three topics from the theory of these flag domains: 1. Measurability(i.e. existence of $G_\mathbb{R}$-invariant Berezinian densities) 2. Global holomorphic superfunctions 3. Cycle spaces and the Double Fibration Transform A revision of the respective classical results is included in the second chapter. This thesis provides a classification of all measurable flag domains and of the global holomorphic functions on all flag domains. Moreove…
Let G be a reductive connected algebraic group over the field of complex numbers. Through the geometric Satake equivalence, the fundamental classes of the Mirković-Vilonen cycles define a basis in each tensor product of rational irreducible representations of G. In the case G=SL_2(C), we show that this basis coincides with the dual canonical basis at q=1.
We show that for the reductive Tannaka groups of semisimple holonomic $\mathscr{D}$-modules on abelian varieties, every Weyl group orbit of weights of their universal cover is realized by a conic Lagrangian cycle on the cotangent bundle. Applications include a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties and a criterion for the simplicity of the arising Lie algebras.
In this article I describe my recent geometric localization argument dealing with actions of NONcompact groups which provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed in [Sch]. A corresponding problem in the compact group setting was solved by N.Berline, E.Getzler and M.Vergne in [BGV] by an application of the theory of equivariant forms and, particularly, the fixed point integral localization formula. This localization argument seems to be the first successful attempt in the direction of building a similar the…
Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map, and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying ou…
The theory of MV cycles associated to a complex reductive group $G$ has proven to be a rich source of structures related to representation theory. We investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac-Moody group. We prove an explicit formula for the Braverman-Finkelberg-Gaitsgory crystal structure on double MV cycles, generalizing a finite-dimensional result of Baumann and Gaussent. As an application, we give a geometric construction of the Naito-Sagaki-Saito crystal via the action of $\hat{SL}_n$ on Fermionic Fock space. In particular, this constructi…
Let g be a complex semisimple Lie algebra, and f : g --> g/G the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of f. We give a generalization of Springer theory to visible, polar representations. It is a class of rational representations of complex reductive groups, for which the invariant theory works by analogy with the adjoint representations. Let G|V be such a representation, f : V --> V/G the quotient map, and P the sheaf of nearby cycles of f. We show that the Fourier transform of P is an intersection homology sheaf on V…
We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension. As an application, we give a description of the nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models.
This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori-Hecke algebras. This is commonly referred to as Kottwitz's Conjecture. The reductive groups associated to the PEL local models under consideration are unramified unitary similitude groups with even di…
Let $V$ be an irreducible representation of group $GL_n({\mathbb C})$, which appears as a submodule in $({\mathbb C}^n)^{\otimes d}$, where ${\mathbb C}^n$ is the tautological $n$-dimensional representation of $GL_n$, and $d$ is a non-negative integer. On the one hand, following refs [Gi] and [BG] one can produce a basis in $V$ using irreducible components of Sringer fibers over a nilpotent matrix in ${\mathfrak {gl}}_d$, whose Jordan blocks correspond to the highest weight of $V$. On the other hand, one can produce a basis in $V$ by Mirković-Vilonen cycles, a construction that works for an ar…
