Character formulas for supercuspidal representations of \(GL_{\ell}\), \(\ell\) a prime.

*(English)*Zbl 0618.22006This paper gives a method for computing the characters of the irreducible supercuspidal representations of the group \(G=GL_{\ell}\) over a non- Archimedean local field F of any characteristic, for \(\ell\) a prime. Initially, the characters of smooth irreducible representations \(\pi\) are defined as distributions on G. Results of Harish-Chandra show that this distribution is given by a locally constant function \(\chi_{\pi}\) on the regular set, and if the characteristic of F is 0, by a locally integrable function on G. For \(\ell\) prime, it is known that all irreducible supercuspical representations \(\pi\) of G may be irreducibly induced from one of the two (conjugacy classes of) maximal compact-mod- center subgroups K. In this case, Harish-Chandra’s expression for the character of \(\pi\) (as a distribution) in terms of an orbital integral of a matrix coefficient becomes the Frobenius formula, which involves the character of the inducing representation \(\sigma\).

Work of Carayol on the construction of the inducing representations is reviewed, in the case that K is the normalizer of an Iwahori subgroup, i.e., the ramified case. Results on lattice flags are reviewed and the notion of K-generic element is introduced. The geometry of conjugacy classes in G is investigated and some results on the support of the inducing character are given.

A convergence formula is proved in the present paper, which allows one to interchange summation and integration in the Frobenius formula, with no assumption on the characteristic of F. The computation of \(\chi_{\pi}\) is reduced to the computation of \(\chi_{\sigma}\) on the set of K- generic elements, and of a series which depends only on the level of the inducing representation \(\sigma\).

A formula is thus obtained for the character as a function on the set of all regular elements of G. In particular, rather precise results are given determining the support of the character in terms of ramified and unramified elliptic elements. If \(\pi\) is a ramified (resp., unramified) supercuspical representation, then the character of \(\pi\) off the ramified (resp., unramified) elliptic set depends only on the level of the inducing representation. Further, a description of the germ expansion at the identity and the neighborhood on which it is valid are found.

Work of Carayol on the construction of the inducing representations is reviewed, in the case that K is the normalizer of an Iwahori subgroup, i.e., the ramified case. Results on lattice flags are reviewed and the notion of K-generic element is introduced. The geometry of conjugacy classes in G is investigated and some results on the support of the inducing character are given.

A convergence formula is proved in the present paper, which allows one to interchange summation and integration in the Frobenius formula, with no assumption on the characteristic of F. The computation of \(\chi_{\pi}\) is reduced to the computation of \(\chi_{\sigma}\) on the set of K- generic elements, and of a series which depends only on the level of the inducing representation \(\sigma\).

A formula is thus obtained for the character as a function on the set of all regular elements of G. In particular, rather precise results are given determining the support of the character in terms of ramified and unramified elliptic elements. If \(\pi\) is a ramified (resp., unramified) supercuspical representation, then the character of \(\pi\) off the ramified (resp., unramified) elliptic set depends only on the level of the inducing representation. Further, a description of the germ expansion at the identity and the neighborhood on which it is valid are found.

Reviewer: C.D.Keys

##### MSC:

22E35 | Analysis on \(p\)-adic Lie groups |

22E50 | Representations of Lie and linear algebraic groups over local fields |