1,721,451 research outputs found
Shalika newforms for GL(n)
Full text linkLet (pi,V) be a generic irreducible representation of a general linear group
over a p-adic field. Jacquet, Piatetski-Shapiro, and Shalika gave an open
compact subgroup K, so that the subspace V^K consisting of v in V fixed by K is
one-dimensional. If pi has a Shalika model Lambda, then we call vectors in
Lambda(V) the Shalika forms of pi, and those in Lambda(V^{K}) the Shalika
newforms. In this article, we give a method to determine all values of the
Shalika newforms on the mirabolic subgroup in the case where pi is
supercuspidal. Using this result, we give another Shalika form with nice
properties, which is not fixed by K in the case where the character defining
the Shalika model is ramified.Comment: 40 page
Recommended from our members
A Formula for Some Shalika Germs
No full textIn this article, for nilpotent orbits in (the Lie algebras of) ramified quasi-split unitary groups with two Jordan blocks, we give the values of their Shalika germs at certain equi-valued elements with half-integral depth previously studied by Hales. These elements are parametrized by hyperelliptic curves defined over the residue field, and the values we obtain can be expressed in terms of Frobenius eigenvalues on the l-adic H^1 of the curves, generalizing previous result of Hales on stable subregular Shalika germs. Using the Shalika germ formulas, we obtain some new results on stability and endoscopic transfer of nilpotent orbital integrals.MathematicsShalika germs; orbital integral
On the non-vanishing of Shalika newvectors at the identity
Full text linkLet π be an irreducible admissible unitary ψ-generic representation of the non-archimedean general linear group GL2n(F), which admits an (η, psi;)-Shalika model Sψη(π). In this paper, we show the nonvanishing of all non-zero Shalika newvectors So ∈ Sψη(π) at the identity matrix g = id ∈ GL2n(F), if η is unramified. This complements the analogous result for Whittaker newvectors, which can be read off the formulae established independently by Miyauchi in [Miy14] and the second named author in [Mat13]
Recommended from our members
Comparison between motivic periods with Shalika periods
Full text linkLet F/F^+ be a quadratic imaginary field extension of a totally real field F^+, and pi cong \tilde{\pi} otimes xi be a cuspidal automorphic representation of GL_n(AA_F) obtained from tilde{pi} by twisting a Hecke character xi. In the case of F^+ = QQ, Michael Harris defined arithmetic automorphic periods for certain tilde{pi} in his Crelle paper 1997, and showed that critical values of automorphic L-functions for pi can be interpreted in terms of these arithmetic automorphic periods. Lin Jie generalized his construction and results to the general totally real field F^+ in her thesis. On the other hand, for certain cuspidal representation Pi of GL_{2n}(F^+), which admits a Shalika model, Grobner and Raghuram related their critical values of L-functions to a non-zero complex number (called Shalika periods). We noticed that the automorphic induction AI(pi) of pi, considered by Harris and Lin, will automatically have a Shalika model, and by comparing common critical values of their identical L-functions, we relate the Shalika periods of AI(pi) with arithmetic automorphic periods of tilde{pi}. In the case F^+=QQ, this comparison will express each arithmetic automorphic period in terms of the corresponding Shalika periods
Motivic integration and the regular Shalika germ.
No full textLet G denote an adjoint semisimple algebraic group defined over the integers. There is an analogy between Haar integration on the p-adic integer valued points of G and motivic integration on the complex power series valued points of G . A result of D. Shelstad from p-adic harmonic analysis states that the regular Shalika germ of such a group is an asymptotically constant function. In the present work it is demonstrated that the complex power series analogue of the regular Shalika germ is similarly an asymptotically constant function. There are three main steps involved in this demonstration. The first step is a description of certain affine schemes, together with morphisms from these schemes to the group. This allows the calculation of the complex power series analogue of the regular Shalika germ via motivic integration over such an affine scheme. The second step is the association of a less complicated shadow scheme to each such affine scheme. This association is achieved by analysis of the coordinate rings of the affine schemes. The third step is the use of the transformation rule from motivic integration. This is used to equate a motivic integral on each affine scheme with one on its associated shadow scheme. It is again used to equate the resulting motivic integrals on two different shadow schemes. This exactly means that the complex power series analogue of the regular Shalika germ is an asymptotically constant function.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/123887/2/3106104.pd
Shalika models and p-adic L-functions
Full text linkGehrmann L. Shalika models and p-adic L-functions. Bielefeld: Universitätsbibliothek Bielefeld; 2014
The metaplectic Casselman-Shalika formula
No full textThis paper studies spherical Whittaker functions for central extensions of reductive groups over local fields. We follow the development of Chinta and Offen to produce a metaplectic Casselman-Shalika formula for tame covers of all unramified groups
On the GL(2n) eigenvariety: branching laws, Shalika families and p-adic L-functions
Full text linkIn this paper, we prove that a GL(2n)-eigenvariety is étale over the (pure) weight space at non-critical Shalika points, and construct multi-variable p-adic L-functions varying over the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori level at p, and give p-adic variation of L-values (of regular algebraic cuspidal automorphic representations of GL(2n) admitting Shalika models) over the whole pure weight space. In the case of GL(4), these results have been used by Loeffler and Zerbes to prove cases of the Bloch–Kato conjecture for GSp(4).Our main innovations are: (a) the introduction and systematic study of ‘Shalika refinements’ of local representations of GL(2n), and evaluation of their attached local twisted zeta integrals; and (b) the p-adic interpolation of representation-theoretic branching laws for GL(n) × GL(n) inside GL(2n). Using (b), we give a construction of multi-variable p-adic functionals on the overconvergent cohomology groups for GL(2n), interpolating the zeta integrals of (a). We exploit the resulting non-vanishing of these functionals to prove our main arithmetic applications
A Casselman–Shalika Formula for the Shalika Model of <i>GL</i><sub><i>n</i></sub>
No full textAbstractThe Casselman–Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of p-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of GLn, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of themethod, that was introduced by Y.Hironaka, and in effect reducemany such problems to straightforward calculations on the group.</jats:p
A probabilistic approach to the Shintani-Casselman-Shalika formula
No full textRecall that Jacquet's Whittaker function for a group , in the non-Archimedean case, is essentially proportional to a character of an irreducible representation of the Langlands dual group - a Schur function in the case of . This statement is known as the Shintani-Casselman-Shalika formula.
In my opinion, Shintani's proof for is remarkably different from the more general proof by Casselman-Shalika. In this talk, I will present a probabilistic proof that is the natural generalisation of Shintani's. It explains the appearance of the Weyl character formula from a reflection principle for random walks.Non UBCUnreviewedAuthor affiliation: Institut de Mathématiques de ToulousePostdoctora
- …
