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.


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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