On the computability of some positive-depth supercuspidal characters near the identity

Abstract

Abstract. This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of p-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which the Harish-Chandra local character expansion holds). We construct a parameter space B (that depends on the group and a real number r > 0) for the set of equivalence classes of the representations of minimal depth r satisfying some additional assumptions. This parameter space is essentially a geometric object defined over Q. Given a non-Archimedean local field K with sufficiently large residual characteristic, the part of the character table near the identity element for G(K) that comes from our class of representations is parameterized by the residue-field points of B. The character values themselves can be recovered by specialization from a constructible motivic exponential function, in the terminology of Cluckers and Loeser in a recent paper. The values of such functions are algorithmically computable. It is in this sense that we show that a large part of the character table of the group G(K) is computable.

Integration in algebraically closed valued fields

“…Let I k = KµRV[k] ∩ I. Let [1] 1 ∈ K + µRV [1] be the isomorphism class of ({1}, id, 1) and [RV >1 ] 1 ∈ K + µRV[1] the isomorphism class of (RV >1 , id, 1). Clearly ([1] 1 , [RV >1 ] 1 ) ∈ µI sp .…”

“…In our context, given the fact that some very complicated integral identities are already motivic (see, for example, [1,2,4]), it is reasonable to expect that many other important kinds of integrals are definable in some first-order languages and hence may be studied motivically. We note that, in their recent paper [9], Hrushovski and Kazhdan have developed a partially first-order method to study adelic structures over curves and, in particular, have obtained a global Poisson summation formula.…”

Uniform analysis on local fields and applications to orbital integrals

“…Another motivation is to enable further applications in the line of e.g. [11,27,6,8,26,18]. In particular, in Section 4, we develop uniform in p bounds for the Fourier transforms of orbital integrals, normalized by the discriminant, discussed in more detail below.…”

Integration in algebraically closed valued fields with sections

“…Let B ⊆ VF n and assume that the VF-dimensions of A, B are n. For any definable bijection φ : A −→ B, we can define the Jacobian transformation Let I, µI be the ideals of the groupifications of K + RV[ * , ·]/ I sp , K + µRV[ * ]/ µI sp . By the same calculations as in [20, §6, §7], we see that, the ideal I is generated by [1] 0 + j and the ideal µI is generated by j µ (see Notation 3.29). Note that j is equal to −[1] 0 = −1 in K RV[ * , ·]/I and hence is not a zero-divisor in K RV[ * , ·] (for otherwise [1] 0 + j would be a zero-divisor in K RV[ * , ·], which is clearly impossible).…”

“…Our motivation for extending the Hrushovski-Kazhdan theory to such expansions is twofold. Firstly, this is to prepare the ground for a plausible theory of motivic characters, especially multiplicative ones, which is something we should have if we are to further the (already far-reaching) application of the theory of motivic integration to, say, geometry and representation theory, as demonstrated, for example, in [1,2,3,12,13]. The use of characters in constructing representations in function spaces is beautifully expounded in the (perhaps a bit old-fashioned but still tremendously insightful) work [9].…”