zbMATH — the first resource for mathematics

Proof of the Deligne-Langlands conjecture for Hecke algebras. (English) Zbl 0613.22004
In this paper, the authors are concerned with a special case of the general Langlands conjecture on irreducible representations of reductive p-adic groups.
Let G be a connected reductive algebraic group over \({\mathbb{C}}\) with simply connected derived group; u be a unipotent element of G; (s,q) be a semi- simple element of \(G\times {\mathbb{C}}^{\times}\) such that \(sus^{-1}=u^ q\). Let \(M(u)=\{(g,q)\in G\times {\mathbb{C}}^{\times}\); \(gug^{-1}=u^ q\}^ a \)closed subgroup of \(G\times {\mathbb{C}}^{\times}\); \(M(u,s)=\{(g,a)\in M(u)\); \(g\in Z(s)\}\) an algebraic subgroup of M(u); \(\bar M(u,s)=M(u,s)/M(u,s)^ 0\), where \(M(u,s)^ 0\) is the connected component of M(u,s) containing the identity. Let M be the smallest algebraic subgroup of \(G\times {\mathbb{C}}^{\times}\) containing (s,q), \(R_ M\) be the Grothendieck group of M and \({\mathbb{R}}_ M=R_ M\otimes {\mathbb{C}}\). Let \({\mathbb{H}}\) be the abstract algebra over \({\mathbb{C}}[q,q^{- 1}]\) called Hecke algebra which has a natural homomorphism into the Iwahori-Matsumoto algebra of the extended affine Weyl group.
For an irreducible complex representation \(\rho\) of \(\bar M(u,s)\), the authors construct the ”standard \({\mathbb{H}}\)-module” \({\mathcal M}_{u,s,q,\rho}=\rho^*\otimes_{{\mathbb{C}}}({\mathbb{C}}_{s,q}\otimes_{{\;bbfR}_ M}K^ M_ 0({\mathcal B}_ u))^{\bar M(u,s)}\), where \({\mathbb{C}}_{s,q}={\mathbb{C}}\) is a natural \({\mathbb{R}}_ M\)-module corresponding to (s,q) and \(K^ M_ 0({\mathcal B}_ u)\) is the K-homology on the variety \({\mathcal B}_ u\) of all Borel subgroups containing u. The procedure is dual to that of the authors’ previous work [ibid. 80, 209- 231 (1985; see the preceding review)] in the sense that here they use equivariant K-homology instead of equivariant K-cohomology. Further, they prove that if q is not a root of unity and \(\rho\) appears in the natural representation of \(\bar M(u,s)\) on \(K_ 0({\mathcal B}^ s_ u)\), where \({\mathcal B}^ s_ u=\{B\in {\mathcal B}_ u\), \(s\in B\}\), then the standard \({\mathbb{H}}\)-module \({\mathcal M}_{u,s,q,\rho}\) has a unique simple quotient which depends only on the G-conjugacy class of \((u,s,\rho)\); conversely, any simple \({\mathbb{H}}\)-module in which q acts as multiplication by q is obtained from some \((u,s,\rho)\) as above, and \((u,s,\rho)\) is uniquely determined by the given \({\mathbb{H}}\)-module up to G-conjugacy. An important role in the proof is played by the analysis of the equivariant K-homology of the space of triples \((u,B,B')\) with \(B,B'\in {\mathcal B}_ u.\)
The authors also give the classification of tempered and square integrable representations of \({\mathbb{H}}\).
Reviewer: E.Abe

22E50 Representations of Lie and linear algebraic groups over local fields
11F33 Congruences for modular and \(p\)-adic modular forms
14G20 Local ground fields in algebraic geometry
Full Text: DOI EuDML
[1] [A] Atiyah, M.F.: Global theory of elliptic operators. Proc. Int. Conf. Funct. Anal. Rel. Topics, Tokyo 1969
[2] [AS] Atiyah, M.F., Segal, G.B.: EquivariantK-theory and completion. J. Differ. Geom.3, 1-18 (1969) · Zbl 0215.24403
[3] [BV] Barbash, D., Vogan, D.: Unipotent representation of complex semisimple groups. Ann. Math.121, 41-110 (1985) · Zbl 0582.22007 · doi:10.2307/1971193
[4] [BFM] Baum, P., Fulton, W., MacPherson, R.: Riemann-Roch and topologicalK-theory for singular varieties. Acta Math.143, 155-192 (1979) · Zbl 0474.14004 · doi:10.1007/BF02392091
[5] [BZ] Bernstein, J., Zelevinskii, A.V.: Induced representations of reductivep-adic groups, I. Ann. Sci. E.N.S.10, 441-472 (1977)
[6] [BS] Beynon, W.M., Spaltenstein, N.: Green functions of finite Chevalley groups of typeE n (n=6, 7, 8). J. Algebra88, 584-614 (1984) · Zbl 0539.20025 · doi:10.1016/0021-8693(84)90084-X
[7] [BB] Bialynicky-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math.98, 480-497 (1973) · Zbl 0275.14007 · doi:10.2307/1970915
[8] [BW] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups and representations of reductive groups. Ann. Math. Stud.94 (1980), Princeton Univ. Press · Zbl 0443.22010
[9] [G] Ginsburg, V.: Lagrangian construction for representations of Hecke algebras. (Preprint 1984) · Zbl 0625.22012
[10] [H] Hironaka, H.: Bimeromorphic smoothing of a complex analytic space. Acta Math. Vietnam.2, (no2), Hanoi 1977 · Zbl 0407.32006
[11] [K] Kasparov, G.G.: Topological invariants of elliptic operators I.,K-homology. Izv. Akad. Nauk. SSSR39, 796-838 (1975) · Zbl 0328.58016
[12] [KL1] Kazhdan, D., Lusztig, G.: A topological approach to Springer’s representations. Adv. Math.38, 222-228 (1980) · Zbl 0458.20035 · doi:10.1016/0001-8708(80)90005-5
[13] [KL2] Kazhdan, D., Lusztig, G.: EquivariantK-theory and representations of Hecke algebras II. Invent. Math.80, 209-231 (1985) · Zbl 0613.22003 · doi:10.1007/BF01388604
[14] [Ko] Kostant, B.: The principal 3-dimensional subgroup and the Betti numbers of a complex Lie group. Am. J. Math.81, 973-1032 (1959) · Zbl 0099.25603 · doi:10.2307/2372999
[15] [La] Langlands, R.P.: Problems in the theory of automorphic forms Lect. Modern Anal. Appl., Lect. Notes Math., vol. 170, pp. 18-86. Berlin-Heidelberg-New York: Springer 1970
[16] [L1] Lusztig, G.: Some examples of square integrable representations of semisimplep-adic groups. Trans. Am. Math. Soc.277, 623-653 (1983) · Zbl 0526.22015
[17] [L2] Lusztig, G.: Cells in affine Weyl groups. In: Algebraic groups and related topics, Hotta, R. (ed.), Advanced Studies in Pure Mathematics, vol. 6. Kinokunia, Tokyo and North-Holland, Amsterdam, 1985
[18] [L3] Lusztig, G.: Character sheavesV. Adv. Math.61, 103-155 (1986) · Zbl 0602.20036 · doi:10.1016/0001-8708(86)90071-X
[19] [L4] Lusztig, G.: EquivariantK-theory and representations of Hecke algebras. Proc. Am. Math. Soc.94, 337-342 (1985) · Zbl 0571.22014
[20] [M] Mostow, G.D.: Fully reducible subgroups of algebraic groups. Am. J. Math.78, 200-221 (1956) · Zbl 0073.01603 · doi:10.2307/2372490
[21] [Se] Segal, G.B.: EquivariantK-theory. Publ. Math. IHES34, 129-151 (1968) · Zbl 0199.26202
[22] [Sh] Shoji, T.: On the Green polynomials of classical groups. Invent. math.74, 239-264 (1983) · Zbl 0525.20027 · doi:10.1007/BF01394315
[23] [Sl] Slodowy, P.: Four lectures on simple groups and singularities. Commun. Math. Inst., Rijksuniv. Utr.11 (1980) · Zbl 0425.22020
[24] [Sn] Snaith, V.: On the K√ľnneth spectral sequence in equivariantK-theory. Proc. Camb. Philos. Soc.72, 167-177 (1972) · Zbl 0238.18005 · doi:10.1017/S0305004100046971
[25] [St] Steinberg, R.: On a theorem of Pittie. Topology14, 173-177 (1975) · Zbl 0318.22010 · doi:10.1016/0040-9383(75)90025-7
[26] [T1] Thomason, R.: AlgebraicK-theory of group scheme actions. Proc. Topol. Conf. in honor J. Moore, Princeton 1983
[27] [T2] Thomason, R.: Comparison of equivariant algebraic and topologicalK-theory. Preprint
[28] [Z1] Zelevinskii, A.V.: Induced representations of reductivep-adic groups. II. On irreducible representations ofGL n. Ann. Sci. E.N.S.13, 165-210 (1980)
[29] [Z2] Zelevinskii, A.V.: Ap-adic analogue of the Kazhdan-Lusztig conjecture. Funkts. Anal. Prilozh.15, 9-21 (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.