##
**Cuspidal geometry of p-adic groups.**
*(English)*
Zbl 0634.22009

Let F be a nonarchimedean local field of characteristic 0, \({\mathbb{G}}\) be a reductive F-group, and \(G={\mathbb{G}}(F)\). Recall that an irreducible representation of a compact group is necessarily finite-dimensional, and that the character is a finite sum of diagonal matrix coefficients. In general, matrix coefficients of irreducible discrete series representations satisfy standard orthogonality relations. From this, one gets orthogonality relations for the characters of the irreducible representations of compact groups.

However, the character of an irreducible infinite-dimensional representation \(\pi\) of a non-compact reductive group G is first a distribution. By theorems of Harish-Chandra [Queen’s Papers Pure Appl. Math. 48, 281-347 (1978; Zbl 0433.22012)], such a distribution is given by integration against a locally integrable function \(\theta_{\pi}\), which is locally constant (analytic for real groups) on the regular set.

One of the main results of this paper is that the \(\theta_{\pi}\) satisfy certain orthogonality relations, for \(\pi\) tempered, involving integration over the elliptic set of G, with respect to a measure arising from the Weyl integration formula. This result is conjectured in Harish- Chandra (notes by van-Dijk) Harmonic analysis on reductive p-adic groups [Lect. Notes Math. 162, (1970; Zbl 0202.411)].

The tools developed are important in their own right. In particular, detailed analysis of various spaces of functions and orbital integrals is developed, which leads to the existence of “pseudo-coefficients”. The book Représentations des groupes réductifs sur un corps local [Herrmann, 1984; Zbl 0544.00007] edited by J. Bernstein, P. Deligne, D. Kazhdan and M.-F. Vignéras, contains the result for GL(n), which is then used in proving a certain correspondence exists between discrete series representations over GL(m,D) and GL(n,F), where D is a central division algebra of F of dimension \(d^ 2\), and \(n=md\). Let R(G) be the Grothendieck group spanned by classes of smooth irreducible representations of G, and \(R_ I(G)\) the subgroup generated by the representations induced from a proper parabolic subgroup. Let \(A(G)\subset C_ c^{\infty}(G)\) be the subspace of functions f such that \(\int_{\Omega}f(\omega)d\omega =0\) for any regular nonelliptic conjugacy class \(\Omega\) of G. Let \(J(G)\subset C_ c^{\infty}(G)\) be the subspace of those f for which every invariant distribution vanishes. The author shows that isomorphisms exist between the spaces \(R(G)/R_ I(G)\cong S\cong A(G)/J(G)\). Here S is a certain space of functions on the elliptic set of G described by Harish-Chandra in terms of germs via the Fourier transform on a Lie algebra. The first isomorphism is induced by restriction of characters to the elliptic set. The second isomorphism is given by the map \(f\mapsto \check f\), where \(f\in A(G)\subset C_ c^{\infty}(G)\) and \v{f}(t)\(=\int_{G}f(xtx^{-1})dx\) for t elliptic is, up to normalization, an orbital integral. Results applied include the theorems of Harish-Chandra, the Trace Paley-Wiener theorem for reductive p-adic groups [see the second review below] and M.-F. Vignéras [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 945-961 (1981; Zbl 0499.22011)]. Thus the space S is described as the image of the subspace A(G) of \(C_ c^{\infty}(G)\). The author then extends the result to the analogous subspace of the Schwartz space of G, which contains all matrix coefficients of the discrete series by the Selberg principle. The above isomorphisms then give the existence of pseudo-coefficients for discrete series representations. The orthogonality relations for tempered representations over the elliptic set are a corollary of this analysis combined with the theorem of Harish-Chandra \(\{\) cf. van Dijk notes\(\}\) that characters of discrete series representations are given by orbital integrals of matrix coefficients on the elliptic set.

A result proved in an appendix is essentially that the characters of tempered admissible representations are dense in the space of all invariant distributions on G. Precisely, it is shown that if \(f\in C_ c^{\infty}(G)\) is such that \(<\pi,f>=0\) for all irreducible representations \(\pi\) of G, then \(\int_{\Omega}f(\omega)d\omega =0\) for any regular conjugacy class \(\Omega\). The proof of this result involves both local harmonic analysis on G and global results, e.g., the simple version of the trace formula. The introduction outlines the results and analysis of the paper, and ends with the conjecture that orbital integrals of functions in A(G) may be expanded on the elliptic set in terms of the characters of those irreducible tempered representations whose class in R(G) is not in the subgroup \(R_ I(G)\).

However, the character of an irreducible infinite-dimensional representation \(\pi\) of a non-compact reductive group G is first a distribution. By theorems of Harish-Chandra [Queen’s Papers Pure Appl. Math. 48, 281-347 (1978; Zbl 0433.22012)], such a distribution is given by integration against a locally integrable function \(\theta_{\pi}\), which is locally constant (analytic for real groups) on the regular set.

One of the main results of this paper is that the \(\theta_{\pi}\) satisfy certain orthogonality relations, for \(\pi\) tempered, involving integration over the elliptic set of G, with respect to a measure arising from the Weyl integration formula. This result is conjectured in Harish- Chandra (notes by van-Dijk) Harmonic analysis on reductive p-adic groups [Lect. Notes Math. 162, (1970; Zbl 0202.411)].

The tools developed are important in their own right. In particular, detailed analysis of various spaces of functions and orbital integrals is developed, which leads to the existence of “pseudo-coefficients”. The book Représentations des groupes réductifs sur un corps local [Herrmann, 1984; Zbl 0544.00007] edited by J. Bernstein, P. Deligne, D. Kazhdan and M.-F. Vignéras, contains the result for GL(n), which is then used in proving a certain correspondence exists between discrete series representations over GL(m,D) and GL(n,F), where D is a central division algebra of F of dimension \(d^ 2\), and \(n=md\). Let R(G) be the Grothendieck group spanned by classes of smooth irreducible representations of G, and \(R_ I(G)\) the subgroup generated by the representations induced from a proper parabolic subgroup. Let \(A(G)\subset C_ c^{\infty}(G)\) be the subspace of functions f such that \(\int_{\Omega}f(\omega)d\omega =0\) for any regular nonelliptic conjugacy class \(\Omega\) of G. Let \(J(G)\subset C_ c^{\infty}(G)\) be the subspace of those f for which every invariant distribution vanishes. The author shows that isomorphisms exist between the spaces \(R(G)/R_ I(G)\cong S\cong A(G)/J(G)\). Here S is a certain space of functions on the elliptic set of G described by Harish-Chandra in terms of germs via the Fourier transform on a Lie algebra. The first isomorphism is induced by restriction of characters to the elliptic set. The second isomorphism is given by the map \(f\mapsto \check f\), where \(f\in A(G)\subset C_ c^{\infty}(G)\) and \v{f}(t)\(=\int_{G}f(xtx^{-1})dx\) for t elliptic is, up to normalization, an orbital integral. Results applied include the theorems of Harish-Chandra, the Trace Paley-Wiener theorem for reductive p-adic groups [see the second review below] and M.-F. Vignéras [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 945-961 (1981; Zbl 0499.22011)]. Thus the space S is described as the image of the subspace A(G) of \(C_ c^{\infty}(G)\). The author then extends the result to the analogous subspace of the Schwartz space of G, which contains all matrix coefficients of the discrete series by the Selberg principle. The above isomorphisms then give the existence of pseudo-coefficients for discrete series representations. The orthogonality relations for tempered representations over the elliptic set are a corollary of this analysis combined with the theorem of Harish-Chandra \(\{\) cf. van Dijk notes\(\}\) that characters of discrete series representations are given by orbital integrals of matrix coefficients on the elliptic set.

A result proved in an appendix is essentially that the characters of tempered admissible representations are dense in the space of all invariant distributions on G. Precisely, it is shown that if \(f\in C_ c^{\infty}(G)\) is such that \(<\pi,f>=0\) for all irreducible representations \(\pi\) of G, then \(\int_{\Omega}f(\omega)d\omega =0\) for any regular conjugacy class \(\Omega\). The proof of this result involves both local harmonic analysis on G and global results, e.g., the simple version of the trace formula. The introduction outlines the results and analysis of the paper, and ends with the conjecture that orbital integrals of functions in A(G) may be expanded on the elliptic set in terms of the characters of those irreducible tempered representations whose class in R(G) is not in the subgroup \(R_ I(G)\).

Reviewer: C.D.Keys

### MSC:

22E35 | Analysis on \(p\)-adic Lie groups |

22E50 | Representations of Lie and linear algebraic groups over local fields |

43A80 | Analysis on other specific Lie groups |

### Keywords:

nonarchimedean local field; reductive F-group; irreducible discrete series representations; orthogonality relations; irreducible representations of compact groups; Weyl integration formula; Harish- Chandra; orbital integrals; Grothendieck group; Fourier transform; Schwartz space; Selberg principle; pseudo-coefficients; tempered representations### References:

[1] | Bernstein, J.; Deligne, P., Le “Centre” de Bernstein (1984), Paris: Hermann, Paris |

[2] | Bernstein, J.; Zelevinsky, Z., Induced representations of reductive p-adic groups I, Ann. Sci. Ec. Norm. Sup., 4e série, 10, 441-472 (1977) · Zbl 0412.22015 |

[3] | Bernstein, J.; Deligne, P.; Kazhdan, D., Trace Paley-Wiener theorem for reductive p-adic groups, J. Analyse Math., 47, 180-192 (1986) · Zbl 0634.22011 · doi:10.1007/BF02792538 |

[4] | Bernstein, J.; Deligne, P.; Kazhdan, D.; Vigneras, M.-F., Représentations des groups réductif sur un corps local (1984), Paris: Hermann, Paris · Zbl 0544.00007 |

[5] | A. Borel and N. Wallach,Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Math. Studies, Princeton Univ. Press, 1980. · Zbl 0443.22010 |

[6] | Deligne, P.; Kazhdan, D.; Vigneras, M.-F., Représentations des algèbres centrales simples p-adiques (1984), Paris: Hermann, Paris · Zbl 0583.22009 |

[7] | Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, Queen’s Papers in Pure and Applied Math., 48, 281-346 (1978) · Zbl 0433.22012 |

[8] | Harish-Chandra,The Plancherel formula for reductive p-adic groups, preprint. · Zbl 0365.22016 |

[9] | Harish-Chandra (notes by G. van Dijk),Harmonic analysis on reductive p-adic groups, Springer Lecture Notes162, 1970. · Zbl 0202.41101 |

[10] | A. J. Silberger,Introduction to harmonic analysis on reductive p-adic group, Princeton Math. Notes23, 1979. · Zbl 0458.22006 |

[11] | Tate, J., The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J., 27, 709-719 (1966) · Zbl 0146.06501 |

[12] | Vignéras, M. F., Caractérisation des intégrales orbital sur un group réductif p-adic, J. Fac. Sci. Univ. Tokyo, 28, 14, 945-961 (1982) · Zbl 0499.22011 |

[13] | Weil, A., Sur certaines groupes d’opérateurs unitaires, Acta Math., 111, 143-211 (1964) · Zbl 0203.03305 · doi:10.1007/BF02391012 |

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