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Motivic proof of a character formula for SL(2). (English) Zbl 1179.22018

Suppose that \(K\) is a nonarchimedean local field of odd residual characteristic, and \(G = \text{SL}_2(K)\). It has been known for some time that the character \(\Theta_\pi\) of an irreducible, supercuspidal representation \(\pi\) of \(G\) can be expressed near the identity in terms of Fourier transforms of certain elliptic orbital integrals. More specifically,
\[ \Theta_\pi (\text{cay}(Y)) = \sum_i c_i \hat\mu_{X_i} \]
for all \(Y \in \text{Lie}(G)\) close enough to zero (in a sense that can be made precise). Here, \(\text{cay}\) denotes the Cayley transform, \(\hat\mu_{X_i}\) is the Fourier transform of the orbital integral associated to the elliptic element \(X_i \in \text{Lie}(G)\), and \(c_i\) is a rational constant. In most cases, the sum only involves one nonzero term, whose coefficient is equal to the formal degree of \(\pi\).
One hopes that the basic objects of harmonic analysis (characters, orbital integrals, the coefficients, etc.) are computable. The point of the present paper is that this is indeed true for the objects appearing in the formula above, at least when \(\pi\) has depth zero. Moreover, one can prove this in a conceptual way by means of motivic integration, a theory that uses methods from algebraic geometry and logic to allow one to work in a field-independent manner.
The paper thus serves as a proof of concept for the program of studying harmonic analysis on \(p\)-adic groups via motivic integration.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
03C10 Quantifier elimination, model completeness, and related topics
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