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Peter-Weyl Iwahori algebras. (English) Zbl 1448.22003
Summary: The Peter-Weyl idempotent \(e_{\mathcal{P}}\) of a parahoric subgroup \(\mathcal{P}\) is the sum of the idempotents of irreducible representations of \(\mathcal{P}\) that have a nonzero Iwahori fixed vector. The convolution algebra associated with \(e_{\mathcal{P}}\) is called a Peter-Weyl Iwahori algebra. We show that any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra. Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algebra have a natural conjugate linear anti-involution \(\star\), and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by \(\bullet\), and the Morita equivalence preserves irreducible and unitary modules for \(\bullet\).
22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
Full Text: DOI
[1] Adams, J., Van Leuween, M., Trapa, P., and Vogan, D., Unitary representations of real reductive groups. 2012. arxiv:1212.2192v2
[2] Barbasch, D. and Ciubotaru, D., Star operations for affine Hecke algebras. 2015. arxiv:1504.04361 · Zbl 1183.22008
[3] Barbasch, D. and Ciubotaru, D., Unitary Equivalences for reductive p-adic groups. Amer. J. Math.135(2013), 1633-1674. · Zbl 1283.22011
[4] Barbasch, D., Ciubotaru, D., and Moy, A., An Euler-Poincaré formula for a depth zero Bernstein projector. Represent. Theory23(2019), 154-187. · Zbl 1423.22017
[5] Barbasch, D. and Moy, A., A unitarity criterion for p-adic groups. Invent. Math.98(1989), no. 1, 19-38. · Zbl 0676.22012
[6] Barbasch, D. and Moy, A., Reduction to real infinitesimal character in Hecke algebras. J. Amer. Math. Soc.6(1993), no. 3, 611-635. · Zbl 0835.22016
[7] Bushnell, C. J. and Kutzko, P. C., Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. Soc.77(1998), 582-634. · Zbl 0911.22014
[8] Ciubotaru, D., Types and unitary representations of reductive p-adic groups. Invent. Math.213(2018), 237-269. · Zbl 1426.22012
[9] , Eisenstein series over finite fields. In: Functional analysis and related fields (Proc. Conf. M. Stone, Univ. Chicago, Chicago, Ill., 1968). Springer, 1970, pp. 76-88. · Zbl 0226.20049
[10] Lam, T. Y., Lectures on modules and rings. , Springer-Verlag, New York, 1999. · Zbl 0911.16001
[11] Lusztig, G., Some examples of square integrable representations of semisimple p-adic groups. Trans. Amer. Math. Soc.277(1983), no. 2, 623-653. · Zbl 0526.22015
[12] Lusztig, G., Affine Hecke algebras and their graded version. J. Amer. Math. Soc.2(1989), 599-635. · Zbl 0715.22020
[13] Moy, A. and Prasad, G., Unrefined minimal K-types for p-adic groups. Invent. Math.116(1994), no. 1-3, 393-408. · Zbl 0804.22008
[14] Moy, A. and Prasad, G., Jacquet functors and unrefined minimal K-types. Comment. Math. Helv.71(1996), no. 1, 98-121. · Zbl 0860.22006
[15] Opdam, E., Harmonic analysis for certain representations of graded Hecke algebras. Acta. Math.175(1995), 75-121. · Zbl 0836.43017
[16] Rieffel, M., Morita Equivalence for [[()[]mml:mi[]()]]𝘊[[()[]/mml:mi[]()]]^⋆-algebras and W^⋆-algebras. J. Pure Appl. Algebra5(1974), 51-96. · Zbl 0295.46099
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