## An Euler-Poincaré formula for a depth zero Bernstein projector

HTML articles powered by AMS MathViewer

- by Dan Barbasch, Dan Ciubotaru and Allen Moy PDF
- Represent. Theory
**23**(2019), 154-187 Request permission

## Abstract:

Work of Bezrukavnikov–Kazhdan–Varshavsky uses an equivariant system of trivial idempotents of Moy–Prasad groups to obtain an Euler–Poincaré formula for the r–depth Bernstein projector. We establish an Euler–Poincaré formula for natural sums of depth zero Bernstein projectors (which is often the projector of a single Bernstein component) in terms of an equivariant system of Peter–Weyl idempotents of parahoric subgroups $\mathscr {G}_{F}$ associated to a block of the reductive quotient $\mathscr {G}_{F}/\mathscr {G}^{+}_{F}$.## References

- J. Bernstein,
*Notes of lectures on Representations of p-adic Groups*, Harvard University, Fall 1992, written by K. E. Rumelhart. - J. N. Bernstein,
*Le “centre” de Bernstein*, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32 (French). Edited by P. Deligne. MR**771671** - Roman Bezrukavnikov, David Kazhdan, and Yakov Varshavsky,
*On the depth $r$ Bernstein projector*, Selecta Math. (N.S.)**22**(2016), no. 4, 2271–2311. MR**3573958**, DOI 10.1007/s00029-016-0278-2 - F. Bruhat and J. Tits,
*Groupes réductifs sur un corps local*, Inst. Hautes Études Sci. Publ. Math.**41**(1972), 5–251 (French). MR**327923**, DOI 10.1007/BF02715544 - F. Bruhat and J. Tits,
*Groupes réductifs sur un corps local*, Inst. Hautes Études Sci. Publ. Math.**41**(1972), 5–251 (French). MR**327923**, DOI 10.1007/BF02715544 - J.-F. Dat,
*Quelques propriétés des idempotents centraux des groupes $p$-adiques*, J. Reine Angew. Math.**554**(2003), 69–103 (French, with English summary). MR**1952169**, DOI 10.1515/crll.2003.009 - Pierre Deligne,
*Le support du caractère d’une représentation supercuspidale*, C. R. Acad. Sci. Paris Sér. A-B**283**(1976), no. 4, Aii, A155–A157 (French, with English summary). MR**425033** - Harish-Chandra,
*Eisenstein series over finite fields*, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 76–88. MR**0457579** - N. Iwahori and H. Matsumoto,
*On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups*, Inst. Hautes Études Sci. Publ. Math.**25**(1965), 5–48. MR**185016**, DOI 10.1007/BF02684396 - Robert E. Kottwitz,
*Tamagawa numbers*, Ann. of Math. (2)**127**(1988), no. 3, 629–646. MR**942522**, DOI 10.2307/2007007 - Allen Moy and Gopal Prasad,
*Unrefined minimal $K$-types for $p$-adic groups*, Invent. Math.**116**(1994), no. 1-3, 393–408. MR**1253198**, DOI 10.1007/BF01231566 - Allen Moy and Gopal Prasad,
*Jacquet functors and unrefined minimal $K$-types*, Comment. Math. Helv.**71**(1996), no. 1, 98–121. MR**1371680**, DOI 10.1007/BF02566411 - Ralf Meyer and Maarten Solleveld,
*Resolutions for representations of reductive $p$-adic groups via their buildings*, J. Reine Angew. Math.**647**(2010), 115–150. MR**2729360**, DOI 10.1515/CRELLE.2010.075 - Peter Schneider and Ulrich Stuhler,
*Representation theory and sheaves on the Bruhat-Tits building*, Inst. Hautes Études Sci. Publ. Math.**85**(1997), 97–191. MR**1471867**, DOI 10.1007/BF02699536 - J. Tits,
*Reductive groups over local fields*, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR**546588**

## Additional Information

**Dan Barbasch**- Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853–0099
- MR Author ID: 30950
- Email: barbasch@math.cornell.edu
**Dan Ciubotaru**- Affiliation: Mathematical Institute, Andrew Wiles Building, University of Oxford, Oxford, OX2 6GG, United Kingdom
- MR Author ID: 754534
- Email: dan.ciubotaru@maths.ox.ac.uk
**Allen Moy**- Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay Road, Hong Kong
- MR Author ID: 127665
- Email: amoy@ust.hk
- Received by editor(s): March 10, 2018
- Received by editor(s) in revised form: February 14, 2019
- Published electronically: March 28, 2019
- Additional Notes: The first author was partly supported by NSA grant H98230-16-1-0006.

The second author was partly supported by United Kingdom EPSRC grant EP/N033922/1.

The third author was partly supported by Hong Kong Research Grants Council grant CERG #603813. - © Copyright 2019 American Mathematical Society
- Journal: Represent. Theory
**23**(2019), 154-187 - MSC (2010): Primary 22E50, 22E35
- DOI: https://doi.org/10.1090/ert/525
- MathSciNet review: 3932569