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Received: 3 March 2017 / Revised: 11 March 2017 / Accepted: 4 April 2017

Proof of the
Broué - Malle - Rouquier Conjecture in Characteristic Zero (After I.
Losev and I. Marin - G. Pfeiffer)

Pavel Etingof Department of Mathematics, Massachusetts Institute of
Technology, Cambridge MA 02139
USA

etingof@math.mit.edu

etingof@math.mit.edu

We explain a proof of the Broué - Malle - Rouquier conjecture on Hecke
algebras of complex reflection groups, stating that the Hecke algebra of a finite
complex reflection group $ W$ is free of rank $ |W|$ over the algebra
of parameters, over a field of characteristic zero. This is based on previous work
of Losev, Marin - Pfeiffer, and Rains and the author.

The goal of this note is to explain a proof of the Broué - Malle - Rouquier conjecture ([Broué et al.1998], p. 178), stating that the Hecke algebra of a finite complex reflection group $ W$ is free of rank $ |W|$ over the algebra of parameters, over a field of characteristic zero. This result is not original - it follows immediately from the results of [Losev2015], [Marin and Pfeiffer2017], and [Etingof and Rains2006], but it does not seem to have been stated explicitly in the literature, so we state and prove it for future reference.We note that there have been a lot of results on this conjecture for particular complex reflection groups, reviewed in [Marin2015], e.g. [Ariki1995], [Ariki and Koike1994], [Marin2012], [Marin2014]; we are not giving the full list of references here.

Let $ V$ be a finite dimensional complex vector space, and $ W\subset GL(V)$ a finite complex reflection group, i.e., $ W$ is generated by complex reflections (elements $ s$ such that $ {\rm rank}(1-s)=1$). Let $ S\subset W$ be the set of reflections, and $ V_{\rm reg}:=V{\setminus} \cup_{s\in S}V^s$. Then by Steinberg's theorem, $ W$ acts freely on $ V_{\rm reg}$. Let $ x\in V_{\rm reg}/W$ be a base point. The braid group of $ W$ is the group $ B_W:=\pi_1(V_{\rm {\bf }reg}/W,x)$. We have a surjective homomorphism $ \pi: B_W\to W$ (corresponding to gluing back the reflection hyperplanes $ V^s$), and $ {\rm Ker}\pi$ is called the pure braid group of $ W$, denoted by $ PB_W$. For each $ s\in S$, let $ T_s\in B_W$ be a path homotopic to a small circle around $ V^s$ (it is defined uniquely up to conjugation). Also let $ n_s$ be the order of $ s$. Then $ T_s^{n_s}\in PB_W$, and by the Seifert - van Kampen theorem, $ PB_W$ is the normal closure of the subgroup of $ B_W$ generated by $ T_s^{n_s}$, $ s\in S$. In other words, $ W$ is the quotient of $ B_W$ by the relations $ T_s^{n_s}=1$, $ s\in S$.

Let $ u_{s,i}$, $ i=1,\ldots,n_s$, be variables such that $ u_{s,i}=u_{t,i}$ if $ s$ is conjugate to $ t$ in $ W$. Let $ R:=\mathbb{Z}[u_{s,i}^{\pm 1}, s\in S, i\in [1,n_s]]$.

This conjecture is currently known for all irreducible complex reflection groups except $ G_{17},\ldots,G_{21}$ (according to the Shephard - Todd classification), and there is a hope that these cases can be proved as well using a sufficiently powerful computer (see [Chavli2016a, Chavli2016b, Marin2015] for more details). Also, it is shown in [Broué et al.1998] that to prove the conjecture, it suffices to show that $ H(W)$ is spanned by $ |W|$ elements.

Our main result is

First assume that $ K={\mathbb{C}}$. It also suffices to assume that $ W$ is irreducible. In this case, possible groups $ W$ are classified by Shephard and Todd ([Shephard1954]). Namely, $ W$ belongs to an infinite series, or $ W$ is one of the exceptional groups $ G_n$, $ 4\le n\le 37$. Among these, $ G_n$ with $ 4\le n\le 22$ are rank $ 2$ groups, while $ G_n$ for $ n\ge 23$ are of rank $ \ge 3$.

The case of the infinite series of groups is well known, see [Ariki1995], [Ariki and Koike1994], [Broué et al.1998]. So it suffices to focus on the exceptional groups. Among these, the result is well known for Coxeter groups, which are $ G_{23}=H_3$, $ G_{28}=F_4$, $ G_{30}=H_4$, $ G_{35}=E_6$, $ G_{36}=E_7$, $ G_{37}=E_8$.

For the groups $ G_n$ for $ n=24,25,26,27,29,31,32,33,34$, the result was established in [Marin and Pfeiffer2017] and references therein, see [Marin2015], Subsection 4.1. Thus, Theorem 1.3 is known (in fact, over any coefficient ring) for all $ W$ except those of rank $ 2$.

In the rank 2 case, the following weak version of Theorem 1.3 was established.

Theorem 1.3 now follows from Theorem 2.1 and the following theorem due to I. Losev.

Now let $ v_1,\ldots,v_r$ be generators of $ H(W)$ over $ R$, and $ e_i,\ldots,e_{|W|}\in H(R)$ be elements defining a basis of $ \mathbb{Q}\otimes_{\mathbb Z}H(W)$ over $ \mathbb{Q} \otimes_{\mathbb{Z}}R$ (they exist by Theorem 1.3). Then $ v_i=\sum_j a_{ij}e_j$ for some $ a_{ij}\in {\mathbb{Q}}\otimes_{\mathbb Z}R$. So for some integer $ D>0$ we have $ Dv_i=\sum_j b_{ij}e_j$, with $ b_{ij}\in R$. Since $ H(W)[1/L]$ is a free $ \mathbb{Z}[1/L]$-module, the same relation holds in $ H(W)[1/L]$. Thus, for $ N=LD$, $ H(W)[1/N]$ is a free $ R[1/N]$-module with basis $ e_1,\ldots,e_{|W|}$. ⬜

- 1. The proof of Theorem 1.3 does not extend to positive characteristic, since the proof of Theorem 2.2 uses complex analysis (the Riemann - Hilbert correspondence).
- 2. The last step of the proof of Theorem 1.3 (Swan's theorem) is really needed for purely aesthetic purposes, to establish the original formulation of the conjecture on the nose. As usual, for practical purposes it is normally sufficient to know only that the algebra $ K\otimes_{\mathbb Z}H(W)$ is a projective $ K\otimes_{\mathbb Z}R$-module. In fact, for most applications, including the ones mentioned in Remark 1.4, already Losev's Theorem 2.2 is sufficient.
- 3. One would like to have a stronger version of Theorem 1.3, giving a set-theoretical splitting $ W\to B_W$ of the homomorphism $ \pi$ whose image is a basis of the Hecke algebra. For instance, when $ W$ is a Coxeter group, then such a splitting is well known and is obtained by taking reduced expressions in the braid group. Such a version is currently available (over any base ring) for all irreducible complex reflection groups except $ G_{17},\ldots,G_{21}$, see [Marin2015], [Chavli2016a], [Chavli2016b].
- 4. Here is an
outline of the proof of Theorem 2.2 given in [Losev2015]. Let $ q=e^{2\pi ic}$, and let
$ \boldsymbol{H}_c(W)$ be the rational Cherednik algebra of $ W$ with
parameter $ c$, [Ginzburg et al.2003]. Let $ M\in {\mathcal O}_c(W)$ be a
module from the category $ \mathcal O$ for this algebra. It is shown in [Ginzburg et al.2003] that the localization of
$ M$ to the set $ \mathfrak{h}^{\rm reg}$ of regular points of the reflection
representation $ \mathfrak{h}$ of $ W$ is a vector bundle on
$ \mathfrak{h}^{\rm reg}$ with a flat connection. So for every $ x\in \mathfrak{h}^{\rm reg}$ we get a
monodromy representation of the braid group $ \pi_1(\mathfrak{h}^{\rm reg}/W)$ on the fiber
$ M_x$, which is shown in [Ginzburg et al.2003] to factor through
$ H_q(W)$. This representation is denoted by $ KZ(M)$, and the
functor $ M\mapsto KZ(M)$ is called the Knizhnik - Zamolodchikov (KZ) functor. It
is shown in [Ginzburg et al.2003] that the representation
$ KZ(M)$ of $ H_q(W)$ factors through a certain quotient
$ H_q'(W)$ of $ H_q(W)$ of dimension $ |W|$. Thus, Theorem
2.2 is
equivalent to the statement that every finite dimensional representation of
$ H_q(W)$ is of the form $ KZ(M)$ for some $ M$.
To show this, let $ \mathfrak{h}^{\rm sr}$ be the complement of the intersections of pairs of distinct reflection hyperplanes in $ \mathfrak{h}$. Take a finite dimensional representation $ V$ of $ H_q(W)$, and let $ N=N_V$ be the vector bundle with a flat connection with regular singularities on $ \mathfrak{h}^{\rm reg}$ corresponding to $ V$ under Deligne's multidimensional Riemann - Hilbert correspondence. One then extends $ N$ to a vector bundle $ \widetilde N$ on $ \mathfrak{h}^{\rm sr}$ compatibly with the $ \boldsymbol{H}_c(W)$-action. One then defines $ M:=\Gamma(\mathfrak{h}^{\rm sr},\widetilde N)$ and shows that $ M\in {\mathcal O}_c(W)$ and $ KZ(M)=V$, as desired.

- 5. Here is an
outline of the proof of Theorem 2.1 given in [Etingof and Rains2006]. For the infinite series of
complex reflection groups the result was proved in [Broué et al.1998]. Thus, let $ W\subset GL_2(\mathbb C)$
be an exceptional complex reflection group of rank $ 2$, of type
$ G_4,\ldots,G_{22}$. Then the intersection of $ W$ with the scalars is a
finite cyclic group generated by an element $ Z$. This element
defines a central element of $ H_q(W)$, which we will also call
$ Z$. Let $ W/\langle Z\rangle=G\subset PGL_2(\mathbb C)=SO_3(\mathbb C)$. Then $ G$ is the group of even
elements in a Coxeter group of type $ A_3$, $ B_3$, or
$ H_3$. Using the theory of length in these Coxeter groups, it is shown
that $ \mathbb{C} \otimes_{\mathbb Z}H(W)$ is generated by $ |G|$ elements as a module over
$ \mathbb{C} \otimes_{\mathbb{Z}}R[Z,Z^{-1}]$. Moreover, taking the determinant of the braid relation of this
algebra in its finite dimensional representations, we find that $ Z^d$ is
an element of $ \mathbb{C}\otimes_{\mathbb Z}R$ for some $ d$. This implies that
$ \mathbb{C}\otimes_{\mathbb Z}H(W)$ is a finite rank module over $ \mathbb{C}\otimes_{\mathbb Z}R$, as desired.
We note that this argument works over an arbitrary base ring. A much more detailed description of this argument is given in [Chavli2016b].

[Ariki1995] Ariki, S.: Representation theory of a Hecke algebra of
$ G(r;p;n)$. J. Algebra 177
, 164--185 (1995)

[Ariki and Koike1994] Ariki, S., Koike, K.: A Hecke algebra of $ ({\mathbb{Z}}/rZ)\wr S_n$ and construction of its irreducible representations. Adv. Math. 106 , 216--243 (1994)

[Broué et al.1998] Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500 , 127--190 (1998)

[Chavli2016a] Chavli, E.: The BMR freeness conjecture for the tetrahedral and octahedral families (2016a). arXiv:1607.07023

[Chavli2016b] Chavli, E.: The Broué-Malle-Rouquier conjecture for the exceptional groups of rank 2. Ph.D. Thesis, U. Paris Diderot (2016b). arXiv:1608.00834

[Deligne and Mostow1993] Deligne, P., Mostow, G.D.: Commensurabilities among Lattices in PU(1, n), Annals of Mathematics Studies 132. Princeton Univ. Press, Princeton (1993)

[Eisenbud1994] Eisenbud, D.: Commutative algebra with a view towards algebraic geometry, Graduate Texts in Mathematics, vol. 150 (1994)

[Etingof and Rains2006] Etingof, P., Rains, E.: Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups. J. Algebra 299 (2), 570--588 (2006)

[Ginzburg et al.2003] Ginzburg, V., Guay, N., Opdam, E., Rouquier, R.: On category O for rational Cherednik algebras. Invent. Math. 154 (3), 617--651 (2003)

[Hartshorne1977] Hartshorne, R.: Algebraic geometry, Graduate texts in mathematics. Springer, Berlin (1977)

[Lam2006] Lam, T.Y.: Serre's problem on Projective modules, Springer monographs in mathematics (2006)

[Losev2015] Losev, I.: Finite-dimensional quotients of Hecke algebras. Algebra Number Theory 9 (2), 493--502 (2015)

[Marin2014] Marin, I.: The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group $ G_{26}$. J. Pure Appl. Algebra 218 , 704--720 (2014)

[Marin2012] Marin, I.: The cubic Hecke algebra on at most 5 strands. J. Pure Appl. Algebra 216 , 2754--2782 (2012)

[Marin2015] Marin, I.: Report of the Broué-Malle-Rouquier conjectures. In: Proceedings of the INDAM intensive period "Perspectives in Lie theory" (2015).

[Marin and Pfeiffer2017] Marin, I., Pfeiffer, G.: The BMR freeness conjecture for the 2-reflection groups. Math. Comput. (2016). arXiv:1411.4760

[Shan2011] Shan, P.: Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras. Annales scientifiques de l'École Normale Supérieure 44 (1), 147---182 (2011)

[Shephard1954] Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6 , 274304 (1954)

[Ariki and Koike1994] Ariki, S., Koike, K.: A Hecke algebra of $ ({\mathbb{Z}}/rZ)\wr S_n$ and construction of its irreducible representations. Adv. Math. 106 , 216--243 (1994)

[Broué et al.1998] Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500 , 127--190 (1998)

[Chavli2016a] Chavli, E.: The BMR freeness conjecture for the tetrahedral and octahedral families (2016a). arXiv:1607.07023

[Chavli2016b] Chavli, E.: The Broué-Malle-Rouquier conjecture for the exceptional groups of rank 2. Ph.D. Thesis, U. Paris Diderot (2016b). arXiv:1608.00834

[Deligne and Mostow1993] Deligne, P., Mostow, G.D.: Commensurabilities among Lattices in PU(1, n), Annals of Mathematics Studies 132. Princeton Univ. Press, Princeton (1993)

[Eisenbud1994] Eisenbud, D.: Commutative algebra with a view towards algebraic geometry, Graduate Texts in Mathematics, vol. 150 (1994)

[Etingof and Rains2006] Etingof, P., Rains, E.: Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups. J. Algebra 299 (2), 570--588 (2006)

[Ginzburg et al.2003] Ginzburg, V., Guay, N., Opdam, E., Rouquier, R.: On category O for rational Cherednik algebras. Invent. Math. 154 (3), 617--651 (2003)

[Hartshorne1977] Hartshorne, R.: Algebraic geometry, Graduate texts in mathematics. Springer, Berlin (1977)

[Lam2006] Lam, T.Y.: Serre's problem on Projective modules, Springer monographs in mathematics (2006)

[Losev2015] Losev, I.: Finite-dimensional quotients of Hecke algebras. Algebra Number Theory 9 (2), 493--502 (2015)

[Marin2014] Marin, I.: The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group $ G_{26}$. J. Pure Appl. Algebra 218 , 704--720 (2014)

[Marin2012] Marin, I.: The cubic Hecke algebra on at most 5 strands. J. Pure Appl. Algebra 216 , 2754--2782 (2012)

[Marin2015] Marin, I.: Report of the Broué-Malle-Rouquier conjectures. In: Proceedings of the INDAM intensive period "Perspectives in Lie theory" (2015).

[Marin and Pfeiffer2017] Marin, I., Pfeiffer, G.: The BMR freeness conjecture for the 2-reflection groups. Math. Comput. (2016). arXiv:1411.4760

[Shan2011] Shan, P.: Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras. Annales scientifiques de l'École Normale Supérieure 44 (1), 147---182 (2011)

[Shephard1954] Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6 , 274304 (1954)

- 99
- × Ariki, S.: Representation theory of a Hecke algebra of $ G(r;p;n)$. J. Algebra 177 , 164--185 (1995)
- × Ariki, S., Koike, K.: A Hecke algebra of $ ({\mathbb{Z}}/rZ)\wr S_n$ and construction of its irreducible representations. Adv. Math. 106 , 216--243 (1994)
- × Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500 , 127--190 (1998)
- × Chavli, E.: The BMR freeness conjecture for the tetrahedral and octahedral families (2016a). arXiv:1607.07023
- × Chavli, E.: The Broué-Malle-Rouquier conjecture for the exceptional groups of rank 2. Ph.D. Thesis, U. Paris Diderot (2016b). arXiv:1608.00834
- × Deligne, P., Mostow, G.D.: Commensurabilities among Lattices in PU(1, n), Annals of Mathematics Studies 132. Princeton Univ. Press, Princeton (1993)
- × Eisenbud, D.: Commutative algebra with a view towards algebraic geometry, Graduate Texts in Mathematics, vol. 150 (1994)
- × Etingof, P., Rains, E.: Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups. J. Algebra 299 (2), 570--588 (2006)
- × Ginzburg, V., Guay, N., Opdam, E., Rouquier, R.: On category O for rational Cherednik algebras. Invent. Math. 154 (3), 617--651 (2003)
- × Hartshorne, R.: Algebraic geometry, Graduate texts in mathematics. Springer, Berlin (1977)
- × Lam, T.Y.: Serre's problem on Projective modules, Springer monographs in mathematics (2006)
- × Losev, I.: Finite-dimensional quotients of Hecke algebras. Algebra Number Theory 9 (2), 493--502 (2015)
- × Marin, I.: The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group $ G_{26}$. J. Pure Appl. Algebra 218 , 704--720 (2014)
- × Marin, I.: The cubic Hecke algebra on at most 5 strands. J. Pure Appl. Algebra 216 , 2754--2782 (2012)
- × Marin, I.: Report of the Broué-Malle-Rouquier conjectures. In: Proceedings of the INDAM intensive period "Perspectives in Lie theory" (2015).
- × Marin, I., Pfeiffer, G.: The BMR freeness conjecture for the 2-reflection groups. Math. Comput. (2016). arXiv:1411.4760
- × Shan, P.: Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras. Annales scientifiques de l'École Normale Supérieure 44 (1), 147---182 (2011)
- × Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6 , 274304 (1954)

Definition 1.1 Conjecture
1.2 Theorem
1.3 Remark
1.4

2. Proof of Theorem 1.3
Theorem 2.1 Theorem
2.2 Corollary
1 Remark
2.3