Depth-zero supercuspidal \(L\)-packets and their stability.

*(English)*Zbl 1193.11111This 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.

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.

Reviewer: Jiuzu Hong (Tel Aviv)

##### 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 |