Let E be a connected reductive algebraic group over C and let W be its Weyl group. The Springer correspondence allows us to parametrize the irreducible representations E of W as F = F^^ where u is a unipotent element in G (up to conjugacy) and <p is an irreducible representation of the group of components AH(u) = ZH(u)IZ°H(u). (However, not all <p arise in the parametrization.) For F = F^Uttp) ...

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In this paper we give a geometric version of the Satake isomorphism [Sat]. As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the coroots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a red...

These are notes for the third meeting of the Atlas of reductive Lie groups project at AIM, in Palo Alto. They describe how to take the description of the representation theory of a real reductive Lie group (cf. Jeff Adams’ notes from last year) to finite combinatorial terms, that can be implemented in a computer. These ideas evolved during my stay at MIT last fall, and benefited immensely from ...

Let F be a finite extension of Qp. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F -split p-adic reductive group over Fp to be supersingular. We then give the classification of irreducible admissible smooth GLn(F )-representations over Fp in terms of supersingular representations. As a consequence we deduce that supersingular...

Consider real reductive group G, as defined in [Wal88]. Let Π be an irreducible admissible representation of G with the distribution character ΘΠ, [Har51]. Denote by uΠ the lowest term in the asymptotic expansion of ΘΠ, [BV80]. This is a finite linear combination of Fourier transforms of nilpotent coadjoint orbits, uΠ = ∑ O cOμ̂O. As shown by Rossmann, [Ros95], the closure of the union of the ni...

A good way to understand an object of study is, as Richard Feynman famously remarked, to “just look at the thing!”. In this paper we apply Feynman’s method to answer the following question: given a surface group representation in Sp(4,R), under what conditions can it be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)? A surface group representation in a...

which is a linear combination of closures of conormal bundles to submanifolds of X. Intuitively, the microlocal multiplicities cα( ) measure the singularity of at α. In settings related to representation theory, a group G acts on X, is G-equivariant, and the microlocal multiplicities play a significant but only partially understood role in representation theory (see [Ro], [SV], [ABV], and [KaSa...