To analyze the standard intertwining operators for induced representations of a reductive group, one is naturally led to study certain integrals over the corresponding unipotent radical N . A Levi component M acts on N by the adjoint action, and so we may decompose the integral according to the Ad(M )-orbits. We treat unitary and classical cases in which M is the product G × H of two groups related by the norm correspondence of Kottwitz-Shelstad. The result is a Weyl integration-type formula for the integral over N . Our expression for the functional illuminates the interaction between the matched conjugacy classes of G and H. This is part of an ongoing project to relate the poles of these operators to twisted endoscopy.Here D is the usual discriminant in H, and D θ is the twisted discriminant via a suitable involution θ. We write W H (T ) for the Weyl group of T in H. The sign ± is '+' in the odd orthogonal case, and '−' in all other cases.Our methods also apply to the case whenG and H are unitary; in this case we have: Result 2. (Unitary Case) Let f ∈ C c (N ), and d m n an Ad(M )-invariant measure on N . Then there is a positive constant c > 0 so that N f (n)d m n = c · T
We show that the residue at s = 0 of the standard intertwining operator attached to a supercuspidal representation π ⊗ χ of the Levi subgroup GL 2 (F ) × E 1 of the quasisplit group SO * 6 (F ) defined by a quadratic extension E/F of p-adic fields is proportional to the pairing of the characters of these representations considered on the graph of the norm map of Kottwitz-Shelstad. Here π is self-dual, and the norm is simply that of Hilbert's theorem 90. The pairing can be carried over to a pairing between the character on E 1 and the character on E × defining the representation of GL 2 (F ) when the central character of the representation is quadratic, but non-trivial, through the character identities of Labesse-Langlands. If the quadratic extension defining the representation on GL 2 (F ) is different from E the residue is then zero. On the other hand when the central character is trivial the residue is never zero. The results agree completely with the theory of twisted endoscopy and L-functions and determines fully the reducibility of corresponding induced representations for all s.
Abstract. We give a closed formula for the number of partitions λ of n such that the corresponding irreducible representation V λ of Sn has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the 2-core towers of such partitions. We also obtain a formula for the number of partitions of n such that the associated permutation representation of Sn has non-trivial determinant.
Let˜GLet˜ Let˜G be a symplectic or even orthogonal group over a p-adic field F , and M the Levi factor of a maximal parabolic subgroup of˜Gof˜ of˜G. Suppose that M has the shape of three blocks of the same size. Let π be a supercuspidal representation of M. In this paper, we give a simple explicit expression for the residue of the standard intertwining operator for the parabolic induction of π from M to G.
Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of extensions of $R[t]$-modules. Using this theory, we describe the similarity classes in $M_n(R)$ for $n\leq 4$, along with their centralizers. Among these, we characterize those classes which are similar to their transposes. Non-self-transpose classes are shown to exist for all $n>3$. When $R$ has finite residue field of order $q$, we enumerate the similarity classes and the cardinalities of their centralizers as polynomials in $q$. Surprisingly, the polynomials representing the number of similarity classes in $M_n(R)$ turn out to have non-negative integer coefficients.Comment: 46 page
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