We determine essentially completely the theta correspondence arising from the dual pair PGL3 × G2 ⊂ E6 over a p-adic field. Our first result determines the theta lift of any non-supercuspidal representation of PGL3 and shows that the lifting respects Langlands functoriality. Our second result shows that the theta lift θ(π) of a (non-self-dual) supercuspidal representation π of PGL3 is an irredu...

We compute the universal deformations of cuspidal representations π of GL2(F ) over Fl, where F is a local field of residue characteristic p and l is an odd prime different from p. When π is supercuspidal there is an irreducible, two dimensional representation ρ of GF that corresponds to π via the mod l local Langlands correspondence of [Vi2]; we show that there is a natural isomorphism between...

Weighted orbital integrals are the terms which occur on the geometric side of the trace formula. We shall investigate these distributions on a p-adic group. We shall evaluate the weighted orbital integral of a supercuspidal matrix coefficient as a multiple of the corresponding character.

For a p-adic field F of characteristic zero, the embeddings of a tame supercuspidal representation π of G = GLn(F ) in the space of smooth functions on the set of symmetric matrices in G are determined. It is shown that the space of such embeddings is nonzero precisely when −1 is in the kernel of π and, in this case, this space has dimension four. In addition, the space of H-invariant linear fo...

Abstract This paper gives a classification of stable vectors in dual Vinberg representations coming from graded Lie algebra type F 4 way that is independent the field definition. Relating these gradings to Moy–Prasad filtrations, we obtain input for Reeder–Yu’s construction epipelagic supercuspidal representations. As corollary, this new $F_{4}(\mathbb {Q}_{p})$ <mml:math xmlns:mml="http://www....

We introduce a new approach to the representation theory of reductive p-adic groups G, based on the Geometric Invariant Theory (GIT) of Moy-Prasad quotients. Stable functionals on these quotients are used to give a new construction of supercuspidal representations of G having small positive depth, called epipelagic. With some restrictions on p, we classify the stable and semistable functionals ...

We construct a Langlands parameterization of supercuspidal representations $G_2$ over $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set generic $G_2(K)$ to irreducible continuous homomorphisms $\rho W_K \to G_2(\CC)$ with $W_K$ Weil group $K$. The construction map is simply matter assembling ...