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Register a new userOn Satake parameters for representations with parahoric fixed vectors
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Authors
Thomas J. Haines
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This article constructs the Satake parameter for any irreducible smooth $J$-spherical representation of a $p$-adic group, where $J$ is any parahoric subgroup. This parametrizes such representations when $J$ is a special maximal parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in formulating a geometric Satake isomorphism for such groups over local function fields.
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We give an explicit construction of test vectors for $T$-equivariant linear functionals on representations $Pi$ of $GL_2$ of a $p$-adic field $F$, where $T$ is a non-split torus. Of particular interest is the case when both the representations are ramified; we completely solve this problem for principal series and Steinberg representations of $GL_2$, as well as for depth zero supercuspidals over $mathbf{Q}_p$. A key ingredient is a theorem of Casselman and Silberger, which allows us to quickly reduce almost all cases to that of the principal series, which can be analyzed directly. Our method shows that the only genuinely difficult cases are the characters of $T$ which occur in the primitive part (or type) of $Pi$ when $Pi$ is supercuspidal. The method to handle the depth zero case is based on modular representation theory, motivated by considerations from Deligne-Lusztig theory and the de Rham cohomology of Deligne-Lusztig-Drinfeld curves. The proof also reveals some interesting features related to the Langlands correspondence in characteristic $p$. We show in particular that the test vector problem has an obstruction in characteristic $p$ beyond the root number criterion of Waldspurger and Tunnell, and exhibits an unexpected dichotomy related to the weights in Serres conjecture and the signs of standard Gauss sums.
We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.
For certain characters of the compact torus of a reductive $p$-adic group, which we call strongly parabolic characters, we prove Satake-type isomorphisms. Our results generalize those of Satake, Howe, Bushnell and Kutzko, and Roche.
Let $G$ be a connected reductive group over the non-archime-dean local field $F$ and let $pi$ be a supercuspidal representation of $G(F)$. The local Langlands conjecture posits that to such a $pi$ can be attached a parameter $L(pi)$, which is an equivalence class of homomorphisms from the Weil-Deligne group with values in the Langlands $L$-group ${}^LG$ over an appropriate algebraically closed field $C$ of characteristic $0$. When $F$ is of positive characteristic $p$ then Genestier and Lafforgue have defined a parameter, $L^{ss}(pi)$, which is a homomorphism $W_F ra {}^LG(C)$ that is {it semisimple} in the sense that, if the image of $L^{ss}(pi)$, intersected with the Langlands dual group $hat{G}(C)$, is contained in a parabolic subgroup $P subset hat{G}(C)$, then it is contained in a Levi subgroup of $P$. If the Frobenius eigenvalues of $L^{ss}(pi)$ are pure in an appropriate sense, then the local Langlands conjecture asserts that the image of $L^{ss}(pi)$ is in fact {it irreducible} -- its image is contained in no proper parabolic $P$. In particular, unless $G = GL(1)$, $L^{ss}(pi)$ is ramified: it is non-trivial on the inertia subgroup $I_F subset W_F$. In this paper we prove, at least when $G$ is split and semisimple, that this is the case provided $pi$ can be obtained as the induction of a representation of a compact open subgroup $U subset G(F)$, and provided the constant field of $F$ is of order greater than $3$. Conjecturally every $pi$ is compactly induced in this sense, and the property was recently proved by Fintzen to be true as long as $p$ does not divide the order of the Weyl group of $G$. The proof is an adaptation of the globalization method of cite{GLo} when the base curve is $PP^1$, and a simple application of Delignes Weil II.
We consider the centers of the affine vertex algebras at the critical level associated with simple Lie algebras. We derive new formulas for generators of the centers in the classical types. We also give a new formula for the Capelli-type determinant for the symplectic Lie algebras and calculate the Harish-Chandra images of the Casimir elements arising from the characteristic polynomial of the matrix of generators of each classical Lie algebra.
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