We show how l-adic sheaves on the rigid analytic Lie algebra for a p-adic group may be used to associate orbital integrals to generalised Green's functions on the reductive quotient of a parahoric.

Beginning with the fundamental work of Allen Moy and Gopal Prasad, the elegant structure theory of Bruhat and Tits has played an increasingly important role in both the representation theory and harmonic analysis of reductive p-adic groups. In this talk, we shall review some of the basic results in the latter category and discuss work that is currently in progress.

Let k be a p-adic field. Let G be the group of k-rational points of a connected reductive group defined over k. Let p be an irreducible admissible representation of G of positive depth. We find a new character expansion of the character c_p, which depends on K-types contained in p. We also determine a G-domain where this expression is valid. This is a joint work with Fiona Murnaghan.

Let G be a split reductive p-adic group. In this talk, we will establish an explicit link between principal nilpotent orbits of G and the irreducible constituents of principal series of G. A geometric characterization of certain irreducible constituents is also provided. In addition, we can express the relation in terms of L-group objects.

We will derive an alternative derivation of the Kudla-Rallis regularized Siegel-Weil formula, using Arthur truncation. We show that a simple criterion determines when the truncated integrals are invariant, and relate it to the assumptions of Kudla-Rallis on the relative sizes of the dual groups.

Suppose F is a p-adic field containing the n-th roots of unity. A metaplectic covering of GL(r,F) is a non-trivial n-fold covering group of GL(n,F). We shall provide a classification of the irreducible unitary representations of these metaplectic coverings, and discuss an application to the automorphic representations of metaplectic coverings.

Let G be the F-rational points of a connected reductive F-group, where F is a p-adic field. Let H be the fixed points of an involution of G. A representation of G is said to be H-distinguished whenever there exists a nonzero H-invariant linear functional on the space of the representation. We will discuss distinguishedness of tame supercuspidal representations of G in terms of the inducing data defined by J.-K. Yu.

The principal series representations of p-adic SL(2,k) are restricted to SL(2,O), where O denotes the integer ring in k. This subgroup represents one of the two conjugacy classes of maximal compact subgroups in SL(2,k). The decomposition of these restricted representations is described in terms of Shalika's classification by orbits of irreducible representations of SL(2,O).

The maximal tori and normal triples that I shall describe in this talk arise naturally in the study of nilpotent orbits of Lie groups and play an important role in several problems such as: classification of nilpotent orbits of real Lie groups, description of admissible nilpotent orbits of real Lie groups, classification of spherical nilpotent orbits, determination of component groups of centralizers of nilpotents in symmetric spaces. I shall present a simple algorithm for computing such tori and discuss two of the above applications.

We study an equivalence relation on the set of ideals in the nilradical of a Borel subalgebra. This appears to be related to the Springer correspondence. It also has a connection with Kazhdan-Lusztig cells in the affine Weyl group and we will explain a theorem concerning this connection.

Let O be the ring of integers of a local field, with maximal ideal P. Write Sp_2n(R) for the symplectic group of rank 2n over the quotient ring R=O/P^l. The Weil representation W of Sp_2n(R) is defined, its irreducible constituents are determined, their Clifford theory is elucidated, and their character fields and Schur indices are computed. A character formula for the restriction of W to the unitary group U_n([`(R)]), [`(R)] a quadratic extension of R, is given.

Let p be a generic discrete representation of a Levi subgroup of SO(2n+1)(F), where F is a p-adic field. Then the R-group of Arthur and the classical R-group of the Aubert involution of p are isomorphic. This is a join work with Ban.