## Depth-zero supercuspidal $$L$$-packets and their stability.(English)Zbl 1193.11111

This is a paper on verifying the local Langlands correspondence for pure inner forms of unramified $$p$$-adic groups and tame Langlands parameters in “general position”. Given each tame Langlands parameter, the authors construct, in a natural way, an $$L$$-packet of depth-zero supercuspidal representation of the appropriate $$p$$-adic group, and verify some expected properties of this $$L$$-packet. In particular, they check the stability of their $$L$$-packets, i.e., they prove that, with some conditions on the base field, the appropriate sum of characters of the representations in the $$L$$-packet is stable, and no proper subset of their $$L$$-packets can form a stable combination.
In the spirit of local class field theory, the authors construct both the “geometric” and “$$p$$-adic” sides of local Langlands correspondence, and make an explicit connection between the two sides.
Let $$k$$ be a $$p$$-adic field of characteristic zero, and $$G$$ be a reductive group over $$k$$. Let $$K$$ be the maximal unramified extension of $$k$$. Assume $$G$$ is quasi-split over $$k$$, and split over $$K$$. Let $$F$$ be a Frobenius element in $$\text{Gal}(K/k)$$. Then it induces an action on a root system, and hence induces an automorphism $$\sigma$$ on the Langlands dual group $$G^\vee$$. The authors start with the geometric side, from a Langlands parameter which is a continuous homomorphism from tame Galois group to the semiproduct of $$G^\vee$$ and $$\langle\sigma\rangle$$, they construct an $$L$$-packet, which is a finite set of irreducible representations of $$p$$-adic groups twisted by different inner forms which are induced from data on the Galois side.
On the $$p$$-adic side, take a pair $$(S,\theta)$$, where $$S=S(K)$$ is the set of $$K$$-points in an unramified $$k$$-anisotropic maximal torus $$S$$ in $$G$$, and $$\theta$$ is a depth-zero character of $$S^F$$, where $$F$$ is the Frobenius. From such data, one can construct a class function $$R(G,S,\theta)$$ on the set of regular semisimple elements of $$G^F$$, which depends only on the $$G^F$$-orbits of the pair $$(S,\theta)$$. For every $$G$$-stable orbit $$T$$ of $$(S, \theta)$$, it is a finite disjoint union of $$G^F$$-orbits, so one can construct a class function $$R(G,T)$$ to be the sum of class functions $$R(G,T_i)$$ ranging over finite $$G^F$$-orbits $$T_i$$ in $$T$$. By comparison between the geometric and $$p$$-adic sides, they prove that the sum of characters of representations in their $$L$$-packets is a function of the form $$R(G,T)$$ for some $$T$$, up to a constant. So the stability of their $$L$$-packets is reduced to show that $$R(G,T)$$ is a stable class-function on the set of strongly regular semisimple elements in $$G^F$$.
One highlight in this paper is that the authors invoke a deep result of Waldspurger to the effect that the fundamental lemma is valid for inner forms which completes their proof.

### MSC:

 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations
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