![Page 1: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/1.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
A Soergel-like category for complexreflection groups of rank one
Thomas Gobet(joint with Anne-Laure Thiel)
University of Sydney
AustMS Annual Conference,University of Adelaide, 7th December 2018.
![Page 2: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/2.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
![Page 3: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/3.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
◮ Let (W,S) be a Coxeter system.
![Page 4: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/4.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
◮ Let (W,S) be a Coxeter system.
◮ Let V be a real reflection faithful representation of(W,S).
![Page 5: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/5.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
◮ Let (W,S) be a Coxeter system.
◮ Let V be a real reflection faithful representation of(W,S). Let R = O(V ) ∼= S(V ∗). It is graded (we setdeg(V ∗) = 2) and W acts degreewise on R.
![Page 6: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/6.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
◮ Let (W,S) be a Coxeter system.
◮ Let V be a real reflection faithful representation of(W,S). Let R = O(V ) ∼= S(V ∗). It is graded (we setdeg(V ∗) = 2) and W acts degreewise on R. For everys ∈ S, set
Bs := R⊗Rs R.
It is an (indecomposable) graded R-bimodule.
![Page 7: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/7.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
◮ Let (W,S) be a Coxeter system.
◮ Let V be a real reflection faithful representation of(W,S). Let R = O(V ) ∼= S(V ∗). It is graded (we setdeg(V ∗) = 2) and W acts degreewise on R. For everys ∈ S, set
Bs := R⊗Rs R.
It is an (indecomposable) graded R-bimodule. Thecategory of graded R-bimodules is Krull-Schmidt.
![Page 8: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/8.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
◮ Let (W,S) be a Coxeter system.
◮ Let V be a real reflection faithful representation of(W,S). Let R = O(V ) ∼= S(V ∗). It is graded (we setdeg(V ∗) = 2) and W acts degreewise on R. For everys ∈ S, set
Bs := R⊗Rs R.
It is an (indecomposable) graded R-bimodule. Thecategory of graded R-bimodules is Krull-Schmidt.
◮ Let x ∈ W . Set Gr(x) := {(xv, v) | v ∈ V } ⊆ V × V .
![Page 9: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/9.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Coxeter groups and Soergel bimodules
◮ Let (W,S) be a Coxeter system.
◮ Let V be a real reflection faithful representation of(W,S). Let R = O(V ) ∼= S(V ∗). It is graded (we setdeg(V ∗) = 2) and W acts degreewise on R. For everys ∈ S, set
Bs := R⊗Rs R.
It is an (indecomposable) graded R-bimodule. Thecategory of graded R-bimodules is Krull-Schmidt.
◮ Let x ∈ W . Set Gr(x) := {(xv, v) | v ∈ V } ⊆ V × V .Then
Bs∼= O(Gr(e) ∪Gr(s)) =: O(e, s).
![Page 10: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/10.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Categorification of the Hecke algebra
![Page 11: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/11.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Categorification of the Hecke algebra
Theorem (Soergel’s categorification theorem)
The (additive, monoidal ⊗R, Karoubian, stable by gradingshifts) category B generated by the Bs, s ∈ S is acategorification of the Iwahori-Hecke algebra H(W ) of W ,
![Page 12: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/12.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Categorification of the Hecke algebra
Theorem (Soergel’s categorification theorem)
The (additive, monoidal ⊗R, Karoubian, stable by gradingshifts) category B generated by the Bs, s ∈ S is acategorification of the Iwahori-Hecke algebra H(W ) of W ,i.e., we have H(W ) ∼= 〈B〉, where 〈B〉 is the splitGrothendieck ring of B.
![Page 13: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/13.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Categorification of the Hecke algebra
Theorem (Soergel’s categorification theorem)
The (additive, monoidal ⊗R, Karoubian, stable by gradingshifts) category B generated by the Bs, s ∈ S is acategorification of the Iwahori-Hecke algebra H(W ) of W ,i.e., we have H(W ) ∼= 〈B〉, where 〈B〉 is the splitGrothendieck ring of B.
◮ Soergel conjectured that the indecomposable objects inB correspond (up to shifts) to Kazhdan and Lusztig’sbasis {C ′
w}w∈W .
![Page 14: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/14.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Categorification of the Hecke algebra
Theorem (Soergel’s categorification theorem)
The (additive, monoidal ⊗R, Karoubian, stable by gradingshifts) category B generated by the Bs, s ∈ S is acategorification of the Iwahori-Hecke algebra H(W ) of W ,i.e., we have H(W ) ∼= 〈B〉, where 〈B〉 is the splitGrothendieck ring of B.
◮ Soergel conjectured that the indecomposable objects inB correspond (up to shifts) to Kazhdan and Lusztig’sbasis {C ′
w}w∈W . This was proven by Elias andWilliamson.
![Page 15: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/15.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Categorification of the Hecke algebra
Theorem (Soergel’s categorification theorem)
The (additive, monoidal ⊗R, Karoubian, stable by gradingshifts) category B generated by the Bs, s ∈ S is acategorification of the Iwahori-Hecke algebra H(W ) of W ,i.e., we have H(W ) ∼= 〈B〉, where 〈B〉 is the splitGrothendieck ring of B.
◮ Soergel conjectured that the indecomposable objects inB correspond (up to shifts) to Kazhdan and Lusztig’sbasis {C ′
w}w∈W . This was proven by Elias andWilliamson. It implies that KL polynomials (forarbitrary Coxeter groups) have nonnegative coefficients,and other important conjectures in rep. theory.
![Page 16: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/16.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Categorification of the Hecke algebra
Theorem (Soergel’s categorification theorem)
The (additive, monoidal ⊗R, Karoubian, stable by gradingshifts) category B generated by the Bs, s ∈ S is acategorification of the Iwahori-Hecke algebra H(W ) of W ,i.e., we have H(W ) ∼= 〈B〉, where 〈B〉 is the splitGrothendieck ring of B.
◮ Soergel conjectured that the indecomposable objects inB correspond (up to shifts) to Kazhdan and Lusztig’sbasis {C ′
w}w∈W . This was proven by Elias andWilliamson. It implies that KL polynomials (forarbitrary Coxeter groups) have nonnegative coefficients,and other important conjectures in rep. theory.
◮ The bounded homotopy category of B can be used toproduce a categorical action of the (generalized) braidgroup of (W,S) (Rouquier).
![Page 17: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/17.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Soergel bimodules for CRG ?
![Page 18: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/18.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Soergel bimodules for CRG ?
◮ There have been attempts to generalize somerepresentation-theoretic aspects of finite reductivegroups to (non-existing) reductive groups attached tocomplex reflection groups (”Spetses” program).
![Page 19: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/19.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Soergel bimodules for CRG ?
◮ There have been attempts to generalize somerepresentation-theoretic aspects of finite reductivegroups to (non-existing) reductive groups attached tocomplex reflection groups (”Spetses” program).Complex reflection groups also have a generalized braidgroup attached to them.
![Page 20: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/20.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Soergel bimodules for CRG ?
◮ There have been attempts to generalize somerepresentation-theoretic aspects of finite reductivegroups to (non-existing) reductive groups attached tocomplex reflection groups (”Spetses” program).Complex reflection groups also have a generalized braidgroup attached to them. Defining Soergel bimodules forthese groups might be useful in the study of theseobjects.
![Page 21: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/21.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Soergel bimodules for CRG ?
◮ There have been attempts to generalize somerepresentation-theoretic aspects of finite reductivegroups to (non-existing) reductive groups attached tocomplex reflection groups (”Spetses” program).Complex reflection groups also have a generalized braidgroup attached to them. Defining Soergel bimodules forthese groups might be useful in the study of theseobjects.
◮ Question: can we construct an analogue of a categoryof Soergel bimodules for CRG ?
![Page 22: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/22.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Soergel bimodules for CRG ?
◮ There have been attempts to generalize somerepresentation-theoretic aspects of finite reductivegroups to (non-existing) reductive groups attached tocomplex reflection groups (”Spetses” program).Complex reflection groups also have a generalized braidgroup attached to them. Defining Soergel bimodules forthese groups might be useful in the study of theseobjects.
◮ Question: can we construct an analogue of a categoryof Soergel bimodules for CRG ?
◮ The definition of Bs still makes sense for a complexreflection group: for V , take the natural module.
![Page 23: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/23.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Soergel bimodules for CRG ?
◮ There have been attempts to generalize somerepresentation-theoretic aspects of finite reductivegroups to (non-existing) reductive groups attached tocomplex reflection groups (”Spetses” program).Complex reflection groups also have a generalized braidgroup attached to them. Defining Soergel bimodules forthese groups might be useful in the study of theseobjects.
◮ Question: can we construct an analogue of a categoryof Soergel bimodules for CRG ?
◮ The definition of Bs still makes sense for a complexreflection group: for V , take the natural module.
◮ First basic observation: if W is a CRG ands ∈ Ref(W ), then if o(s) 6= 2, we have
Bs 6∼= O(e, s).
![Page 24: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/24.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
A Soergel-like category on the rank one case
![Page 25: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/25.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
A Soergel-like category on the rank one case
◮ Let s : C −→ C be a reflection of order d > 2, that is, sis given by multiplication by a primitive d-th root ofunity ζ.
![Page 26: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/26.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
A Soergel-like category on the rank one case
◮ Let s : C −→ C be a reflection of order d > 2, that is, sis given by multiplication by a primitive d-th root ofunity ζ.
◮ Let V = C, R = O(V ), and Gr(x) be as above.
![Page 27: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/27.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
A Soergel-like category on the rank one case
◮ Let s : C −→ C be a reflection of order d > 2, that is, sis given by multiplication by a primitive d-th root ofunity ζ.
◮ Let V = C, R = O(V ), and Gr(x) be as above.
◮ Given A ⊆ W , we write
O(A) := O(
⋃
x∈A
Gr(x))
.
It is a graded, indecomposable R-bimodule.
![Page 28: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/28.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
A Soergel-like category on the rank one case
◮ Let s : C −→ C be a reflection of order d > 2, that is, sis given by multiplication by a primitive d-th root ofunity ζ.
◮ Let V = C, R = O(V ), and Gr(x) be as above.
◮ Given A ⊆ W , we write
O(A) := O(
⋃
x∈A
Gr(x))
.
It is a graded, indecomposable R-bimodule.
Definition
A subset A ⊆ W is cyclically connected ifA = {si, si+1, . . . , si+j} for some 0 ≤ i, j ≤ d− 1.
![Page 29: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/29.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
A Soergel-like category on the rank one case
◮ Let s : C −→ C be a reflection of order d > 2, that is, sis given by multiplication by a primitive d-th root ofunity ζ.
◮ Let V = C, R = O(V ), and Gr(x) be as above.
◮ Given A ⊆ W , we write
O(A) := O(
⋃
x∈A
Gr(x))
.
It is a graded, indecomposable R-bimodule.
Definition
A subset A ⊆ W is cyclically connected ifA = {si, si+1, . . . , si+j} for some 0 ≤ i, j ≤ d− 1.
◮ Let BW be the (additive, graded, monoidal, Karoubian)category generated by Bs := O(e, s).
![Page 30: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/30.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Description of the Grothendieck ring
![Page 31: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/31.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Description of the Grothendieck ring
Theorem (Structure of the Grothendieck ring of BW )
![Page 32: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/32.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Description of the Grothendieck ring
Theorem (Structure of the Grothendieck ring of BW )
1. The indecomposable objects in BW are given by
{O(A) | A ⊆ W cyc. connected}/isoms. and grading shifts.
In particular, the split Grothendieck ring AW := 〈BW 〉 is afree Z[v, v−1]–module of rank d(d− 1) + 1.
![Page 33: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/33.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Description of the Grothendieck ring
Theorem (Structure of the Grothendieck ring of BW )
1. The indecomposable objects in BW are given by
{O(A) | A ⊆ W cyc. connected}/isoms. and grading shifts.
In particular, the split Grothendieck ring AW := 〈BW 〉 is afree Z[v, v−1]–module of rank d(d− 1) + 1.
2. The algebra AW has generators s, C1, · · · , Cd−1 and rel.
![Page 34: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/34.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Description of the Grothendieck ring
Theorem (Structure of the Grothendieck ring of BW )
1. The indecomposable objects in BW are given by
{O(A) | A ⊆ W cyc. connected}/isoms. and grading shifts.
In particular, the split Grothendieck ring AW := 〈BW 〉 is afree Z[v, v−1]–module of rank d(d− 1) + 1.
2. The algebra AW has generators s, C1, · · · , Cd−1 and rel.
sd = 1,
CiCj = CjCi ∀i, j and sCi = Cis ∀i,
C1Ci = Ci+1 + sCi−1 ∀i = 1, . . . , d− 2,
C1Cd−1 =(
v + v−1)
Cd−1,
sCd−1 = Cd−1,
![Page 35: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/35.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
About the proof
![Page 36: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/36.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
About the proof
![Page 37: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/37.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
About the proof, (slightly) more seriously
![Page 38: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/38.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
About the proof, (slightly) more seriously
◮ An important step in the proof of the Theorem above isgiven by
![Page 39: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/39.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
About the proof, (slightly) more seriously
◮ An important step in the proof of the Theorem above isgiven by
Lemma (”Soergel’s Lemma”)
Let A ⊆ W be cyclically connected.
![Page 40: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/40.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
About the proof, (slightly) more seriously
◮ An important step in the proof of the Theorem above isgiven by
Lemma (”Soergel’s Lemma”)
Let A ⊆ W be cyclically connected. There is anisomorphism of graded R-bimodules
O(e, s)⊗R O(A) ∼= O(A ∪ sA)⊕O(A ∩ sA)[−2].
![Page 41: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/41.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
About the proof, (slightly) more seriously
◮ An important step in the proof of the Theorem above isgiven by
Lemma (”Soergel’s Lemma”)
Let A ⊆ W be cyclically connected. There is anisomorphism of graded R-bimodules
O(e, s)⊗R O(A) ∼= O(A ∪ sA)⊕O(A ∩ sA)[−2].
◮ This allows one to decompose tensor powers of O(e, s)in direct sums of O(A), where A ⊆ W is cyclicallyconnected.
![Page 42: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/42.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Semisimplicity
![Page 43: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/43.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Semisimplicity
◮ The algebra AW is an extension of (a version of) theHecke algebra of W (put s = 1 in the abovepresentation).
![Page 44: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/44.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Semisimplicity
◮ The algebra AW is an extension of (a version of) theHecke algebra of W (put s = 1 in the abovepresentation). The Hecke algebra of W is known to besemisimple.
![Page 45: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/45.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Semisimplicity
◮ The algebra AW is an extension of (a version of) theHecke algebra of W (put s = 1 in the abovepresentation). The Hecke algebra of W is known to besemisimple.
◮ Let ACW be the C-algebra defined by the same
presentation as the one in the main theorem, but overC, with v ∈ C
×.
![Page 46: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/46.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Semisimplicity
◮ The algebra AW is an extension of (a version of) theHecke algebra of W (put s = 1 in the abovepresentation). The Hecke algebra of W is known to besemisimple.
◮ Let ACW be the C-algebra defined by the same
presentation as the one in the main theorem, but overC, with v ∈ C
×.
Theorem (Semisimplicity)
If v + v−1 6= 2cos(kπd) for all k = 1, . . . , d− 1, then AC
W is asemisimple algebra.
![Page 47: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/47.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Extended Soergel categories
![Page 48: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/48.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Extended Soergel categories
◮ If d = 2 (type A1), then Soergel’s category has twoindecomposables.
![Page 49: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/49.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Extended Soergel categories
◮ If d = 2 (type A1), then Soergel’s category has twoindecomposables. The bimodule O(s) does not appearas a direct summand of a tensor power of Bs.
![Page 50: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/50.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Extended Soergel categories
◮ If d = 2 (type A1), then Soergel’s category has twoindecomposables. The bimodule O(s) does not appearas a direct summand of a tensor power of Bs. Adding itas generator gives a category whose Grothendieck ringhas the same presentation as the one above.
![Page 51: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/51.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Extended Soergel categories
◮ If d = 2 (type A1), then Soergel’s category has twoindecomposables. The bimodule O(s) does not appearas a direct summand of a tensor power of Bs. Adding itas generator gives a category whose Grothendieck ringhas the same presentation as the one above.
◮ It is unknown what the Grothendieck ring of thecategory generated by Soergel bimodules and O(s),s ∈ S is in the case of Coxeter groups, except in typesA1 and A2.
![Page 52: A Soergel-like category for complex reflection groups of ...gobet/adelaide.pdf · groups of rank one Thomas Gobet (joint with Anne-Laure Thiel) Soergel bimodules Description of the](https://reader033.vdocuments.net/reader033/viewer/2022051603/5ff099d23e18d020323a5682/html5/thumbnails/52.jpg)
A Soergel-likecategory for
complex reflectiongroups of rank one
Thomas Gobet(joint with
Anne-Laure Thiel)
Soergel bimodules
Description of theGrothendieck ring
Semisimplicity
Extended Soergelcategories
Extended Soergel categories
◮ If d = 2 (type A1), then Soergel’s category has twoindecomposables. The bimodule O(s) does not appearas a direct summand of a tensor power of Bs. Adding itas generator gives a category whose Grothendieck ringhas the same presentation as the one above.
◮ It is unknown what the Grothendieck ring of thecategory generated by Soergel bimodules and O(s),s ∈ S is in the case of Coxeter groups, except in typesA1 and A2.
◮ In type A2, this category gives rise to a Grothendieckring of rank 25.