# Affine Hecke algebras and their appearance in mathematics

The goal of this seminar is to understand affine Hecke algebras, their representation theory and their connections to various subfields of mathematics, i.e. to $p$-adic representation theory, geometry and categorification.

### Contact

Catharina Stroppel email
Daniel Tubbenhauer email

### Where and when?

• Usual session:
• Seminarraum, Max-Planck-Institut für Mathematik (MPIM), Vivatsgasse 7, 53111 Bonn, Germany
• Every Monday from 16:15-17:45
• First meeting: 07.11.2016
• Exceptions:
• The talk which was scheduled on Monday the 05.12.2016 will take place Friday the 09.12.2016, 14:15-15:45 in the seminar room 1.008, Mathematik-Zentrum, Endenicher Allee 60
• Question session:
• Seminar room 1.008, Mathematik-Zentrum, Endenicher Allee 60, 53115 Bonn, Germany
• Every second Friday from 14:15-15:45 (if time permits)
• First meeting: If time permits
• Preliminary meeting:
• 24.10.2016, Seminarraum, Max-Planck-Institut für Mathematik (MPIM), Vivatsgasse 7, 53111 Bonn, Germany

### Literature

• [1] I.N. Bernstein and A.V. Zelevinsky, “Induced representations of reductive $p$-adic groups”, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441-472, link: click
• [2] D. Bump, “Hecke algebras”, eprint, link: click
• [3] D. Bump, “Lie groups”, Second edition, Graduate Texts in Mathematics, 225, Springer, New York, 2013
• [4] C.J. Bushnell and G. Henniart, “The local Langlands conjecture for $\mathrm{Gl}(2)$”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Springer-Verlag, Berlin, 2006
• [5] B. Elias, “Quantum Satake in type $A$: part I”, arXiv link: click
• [6] B. Elias, “The two-color Soergel calculus”, Compos. Math. 152 (2016), no. 2, 327-398, arXiv link: click
• [7] B. Elias and G. Williamson, “Soergel calculus”, Represent. Theory 20 (2016), 295-374, arXiv link: click
• [8] R.M. Green, “The affine $q$-Schur algebra”, J. Algebra 215 (1999), no. 2, 379-411, arXiv link: click
• [9] B.H. Gross, “On the Satake isomorphism”, Galois representations in arithmetic algebraic geometry (Durham, 1996), 223-237, London Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press, Cambridge, 1998
• [10] T.J. Haines and R.E. Kottwitz and A. Prasad, “Iwahori-Hecke Algebras”, J. Ramanujan Math. Soc. 25 (2010), no. 2, 113-145, arXiv link: click
• [11] F. Knop, “On the Kazhdan-Lusztig basis of a spherical Hecke algebra”, Represent. Theory 9 (2005), 417-425 (electronic), arXiv link: click
• [12] V. Miemietz and C. Stroppel, “Affine quiver Schur algebras and $p$-adic $\mathrm{Gl}_n$”, arXiv link: click
• [13] A. Ram, “Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux”, Pure Appl. Math. Q. 2 (2006), no. 4, part 2, 963-1013, arXiv link: click
• [14] W. Soergel, “Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules”, Represent. Theory 1 (1997), 83-114 (electronic), link: click

### Schedule

• 07.11.2016, Speaker: Florian Seiffarth, Title: The ordinary Hecke algebras and finite groups
• 14.11.2016, Speaker: Valentin Buciumas, Title: Hecke algebras as convolution algebras and locally profinite groups
• 21.11.2016, Speaker: Tashi Walde, Title: The philosophy of cusp forms
• 28.11.2016, Speaker: Rosanna Laking, Title: Smooth representations of $\mathrm{Gl}(\mathbb{Q}_p)$
• 09.12.2016, Speaker: Matthew Pressland, Title: Induced representations of reductive $p$-adic groups
• 12.12.2016, Speaker: Mikhail Gorsky, Title: The (extended) affine Weyl group and the (extended) affine Hecke algebra
• 19.12.2016, Speaker: Chun-Ju Lai, Title: (Affine) Schur-Weyl duality and $q$-Schur algebras I
• 09.01.2017, Speaker: Leon Barth, Title: (Affine) Schur-Weyl duality and $q$-Schur algebras II
• 16.01.2017, Speaker: Lilit Martirosyan, Title: (Affine) Schur-Weyl duality and $q$-Schur algebras III
• 23.01.2017, Speaker: Deniz Kus, Title: Affine Hecke algebras as alcove walk algebras
• 30.01.2017, Speaker: Emily Norton, Title: The algebraic Satake
• 06.02.2017, Speaker: Hankyung Ko, Title: Kazhdan-Lusztig basis of spherical Hecke algebras
• 13.02.2017, Speaker: Tim Seynnaeve, Title: Categorified Hecke algebras
• 20.02.2017, Speaker: Arik Wilbert, Title: Web calculus and singular Soergel bimodules

### Literature for the talks

• Title: The ordinary Hecke algebras and finite groups, Sources: [2], [3] and [8]
• Title: Hecke algebras as convolution algebras and locally profinite groups, Sources: [2], [3] and [4]
• Title: The philosophy of cusp forms, Sources: [2], [3] and [4]
• Title: Smooth representations of $\mathrm{Gl}(\mathbb{Q}_p)$, Sources: [2], [3] and [4]
• Title: Induced representations of reductive $p$-adic groups, Sources: [1]
• Title: The (extended) affine Weyl group and the (extended) affine Hecke algebra, Sources: [2], [8] and [12]
• Title: (Affine) Schur-Weyl duality and $q$-Schur algebras I, Sources: [8]
• Title: (Affine) Schur-Weyl duality and $q$-Schur algebras II, Sources: [8]
• Title: (Affine) Schur-Weyl duality and $q$-Schur algebras III, Sources: [8]
• Title: Affine Hecke algebras as alcove walk algebras, Sources: [13]
• Title: The algebraic Satake, Sources: [9] and [10]
• Title: Kazhdan-Lusztig basis of spherical Hecke algebras, Sources: [11] and [14]
• Title: Categorified Hecke algebras, Sources: [6] and [7]
• Title: Web calculus and singular Soergel bimodules, Sources: [5] and [6]

### Goals of the talks

• Title: The ordinary Hecke algebras and finite groups,
Goals:
• Definition of Coxeter groups
• Definition of the associated Hecke algebras
• Definition of Hecke algebras for normal subgroups $K < G$ of a finite group $G$

• Title: Hecke algebras as convolution algebras and locally profinite groups,
Goals:
• Construction of finite Hecke algebras
• Bruhat decomposition; general statement and the “simple” proof in case $\mathrm{Gl}_n$, see click
• Connections to the representation theory of the symmetric group, i.e. “$S_n=\mathrm{Gl}_n(\mathbb{F}_1)$”
• Definition of locally profinite groups (for example $\mathbb{Q}_p$)

• Title: The philosophy of cusp forms,
Goals:
• Motivation of the philosophy
• Definition of cuspidal representations of $\mathrm{Gl}_n(\mathbb{F}_q)$
• Introduction of the character ring $R(q)$
• The ring $R(q)$ is commutative
• Explain the “higher” structure in the category lifting properties of the character ring
• “Counting” of cuspidal representations of $\mathrm{Gl}_n(\mathbb{F}_q)$

• Title: Smooth representations of $\mathrm{Gl}(\mathbb{Q}_p)$,
Goals:
• Definition of smooth, admissible representations of $\mathrm{Gl}(\mathbb{Q}_p)$
• Definition of the “global” Hecke algebra
• Definition of the Iwahori subgroup for $\mathrm{Gl}(\mathbb{Q}_p)$
• Classification of irreducible smooth representations of $\mathrm{Gl}(\mathbb{Q}_p)$ via the Iwahori algebra

• Title: Induced representations of reductive $p$-adic groups,
Goals:
• Summary of the paper [1]

• Title: The (extended) affine Weyl group and the (extended) affine Hecke algebra,
Goals:
• Definition of the (extended) affine Weyl group $W_{\mathrm{aff}}$
• Main example is type $\tilde{A}_n$ (infinite permutations, permutations on a cylinder, alcove combinatorics)
• Realization of the affine Weyl group $W_{\mathrm{aff}}=N(T)/T$ for $\mathrm{Gl}(\mathbb{Q}_p)$
• Iwahori(-Matsumoto)-Hecke algebra: definition and first properties

• Title: (Affine) Schur-Weyl duality and $q$-Schur algebras I+II+III,
Goals:
• Explicit description of the Schur algebra
• Statement and proof of Schur-Weyl duality
• Conclusion: Hecke algebras as centralizers
• Applicaiton: The center of the Hecke algebras
• The same as above in the affine case

• Title: Affine Hecke algebras as alcove walk algebras,
Goals:
• Definition of the alcove walk algebras
• Realization of Hecke algebras via alcove walk algebras
• Again, the center of the Hecke algebras

• Title: The algebraic Satake,
Goals:
• State the algebraic Satake
• The Satake isomorphism
• Sketch of the proof of the Satake isomorphism
• State the geometric Satake

• Title: Kazhdan-Lusztig basis of spherical Hecke algebras,
Goals:
• The combinatorics of (affine) Kazhdan-Lusztig polynomials
• The Kazhdan-Lusztig basis
• Its relation to Weyl characters

• Title: Categorified Hecke algebras,
Goals:
• Elias' exotic $\widehat{\mathfrak{sl}_n}$
• From webs to representation of quantum Lie algebras in case $q^\ell=1$