Equivariant cohomology and fixed points of smooth Calogero-Moser spaces

Equivariant cohomology and fixed points of smooth Calogero-Moser spaces

C´edric Bonnaf´e

CNRS (UMR 5149) - Universit´e de Montpellier

Banff, Canada, October 2017

Set-up

Set-up

dimC(V) =n<∞

W <GL_C(V), |W|<∞

Set-up

dimC(V) =n<∞

W <GL_C(V), |W|<∞ cod(w) = codimC(V^w)

cod(ww^′)6cod(w) +cod(w^′)

Set-up

dimC(V) =n<∞

W <GL_C(V), |W|<∞ cod(w) = codimC(V^w)

cod(ww^′)6cod(w) +cod(w^′) Ref(W) =cod⁻¹(1).

Hypothesis: W =hRef(W)i

Set-up

dimC(V) =n<∞

W <GL_C(V), |W|<∞ cod(w) = codimC(V^w)

cod(ww^′)6cod(w) +cod(w^′) Ref(W) =cod⁻¹(1).

Hypothesis: W =hRef(W)i

Set-up (continued)

Set-up (continued)

F_i(CW) =L

cod(w)6iCw

F_i(CW)·F_j(CW)⊂F_i+j(CW)

Set-up (continued)

F_i(CW) =L

cod(w)6iCw

F_i(CW)·F_j(CW)⊂F_i+j(CW)

F₀(CW) = C⊂F₁(CW)⊂F₂(CW)⊂ · · · ⊂F_n(CW) =CW is a filtration ofCW.

Set-up (continued)

F_i(CW) =L

cod(w)6iCw

F_i(CW)·F_j(CW)⊂F_i+j(CW)

F₀(CW) = C⊂F₁(CW)⊂F₂(CW)⊂ · · · ⊂F_n(CW) =CW is a filtration ofCW.

If Ais a subalgebra of CW, we define F_i(A) =A∩F_i(CW)

Set-up (continued)

F_i(CW) =L

cod(w)6iCw

F_i(CW)·F_j(CW)⊂F_i+j(CW)

F₀(CW) = C⊂F₁(CW)⊂F₂(CW)⊂ · · · ⊂F_n(CW) =CW is a filtration ofCW.

If Ais a subalgebra of CW, we define F_i(A) =A∩F_i(CW) ReesF(A) =L

i>0hⁱF_i(A)⊂C[h]⊗A

Set-up (continued)

F_i(CW) =L

cod(w)6iCw

F_i(CW)·F_j(CW)⊂F_i+j(CW)

F₀(CW) = C⊂F₁(CW)⊂F₂(CW)⊂ · · · ⊂F_n(CW) =CW is a filtration ofCW.

If Ais a subalgebra of CW, we define F_i(A) =A∩F_i(CW) ReesF(A) =L

i>0hⁱF_i(A)⊂C[h]⊗A C[h]/hh−ξi ⊗ReesF(A)≃

A if ξ6=0, GradF(A) if ξ=0.

Example: W = S

⊂ GL

( C )

Example: W = S

⊂ GL

( C )

Part(n) ←→^∼ Irr(S_n) λ 7−→ χ_λ

Example: W = S

⊂ GL

( C )

Part(n) ←→^∼ Irr(S_n) λ 7−→ χ_λ

λ7−→

γ_d(λ)∈Part →its d-core λ[d]∈Part_d →its d-quotient

Example: W = S

⊂ GL

( C )

Part(n) ←→^∼ Irr(S_n) λ 7−→ χ_λ

λ7−→

γ_d(λ)∈Part →its d-core λ[d]∈Part_d →its d-quotient

|λ|=|γ_d(λ)|+d |λ[d]|.

Example: W = S

⊂ GL

( C )

Part(n) ←→^∼ Irr(S_n) λ 7−→ χ_λ

λ7−→

γ_d(λ)∈Part →its d-core λ[d]∈Part_d →its d-quotient

|λ|=|γ_d(λ)|+d |λ[d]|.

Theorem (Fong-Srinivasan, 1982)

Assume thato(q mod ℓ) =d. Two unipotent characters ρ_λ and ρ_µ of GL_n(F_q)lie in the same ℓ-block if and only if γ_d(λ) = γ_d(µ).

Example: W = S

⊂ GL

( C ) (continued)

Example: W = S

⊂ GL

( C ) (continued)

Fix a d-coreγ such thatr = n−|γ|

d ∈N

Example: W = S

⊂ GL

( C ) (continued)

Fix a d-coreγ such thatr = n−|γ|

d ∈N S_r ⋉µ^r_d ≃G(d,1,r)⊂GL_r(C)

Example: W = S

⊂ GL

( C ) (continued)

Fix a d-coreγ such thatr = n−|γ|

d ∈N S_r ⋉µ^r_d ≃G(d,1,r)⊂GL_r(C)

Part_d(r) ←→^∼ Irr(G(d,1,r)) µ 7−→ χ^(d)_µ

Example: W = S

⊂ GL

( C ) (continued)

Fix a d-coreγ such thatr = n−|γ|

d ∈N S_r ⋉µ^r_d ≃G(d,1,r)⊂GL_r(C)

Part_d(r) ←→^∼ Irr(G(d,1,r)) µ 7−→ χ^(d)_µ

Let ϕ_γ : Z(CS_n) −→ Z(CG(d,1,r))

e_χλ 7−→

e_χ^(d)

λ[d]

if γ_d(λ) =γ, 0 if γ_d(λ)6=γ,

Example: W = S

⊂ GL

( C ) (continued)

Fix a d-coreγ such thatr = n−|γ|

d ∈N S_r ⋉µ^r_d ≃G(d,1,r)⊂GL_r(C)

Part_d(r) ←→^∼ Irr(G(d,1,r)) µ 7−→ χ^(d)_µ

Let ϕ_γ : Z(CS_n) −→ Z(CG(d,1,r))

e_χλ 7−→

e_χ^(d)

λ[d]

if γ_d(λ) =γ, 0 if γ_d(λ)6=γ,

Theorem (B.-Maksimau-Shan, 2017)

ϕ_γ(F_i(Z(CS_n))⊂F_i(Z(CG(d,1,r)))

Example: W = S

⊂ GL

( C ) (continued)

Fix a d-coreγ such thatr = n−|γ|

d ∈N S_r ⋉µ^r_d ≃G(d,1,r)⊂GL_r(C)

Part_d(r) ←→^∼ Irr(G(d,1,r)) µ 7−→ χ^(d)_µ

Let ϕ_γ : Z(CS_n) −→ Z(CG(d,1,r))

e_χλ 7−→

e_χ^(d)

λ[d]

if γ_d(λ) =γ, 0 if γ_d(λ)6=γ,

Theorem (B.-Maksimau-Shan, 2017)

ϕ_γ(F_i(Z(CS_n))⊂F_i(Z(CG(d,1,r)))

(i.e. (IdC[h]⊗ϕ_γ)(ReesFZ(CS_n))⊂ ReesFZ(CG(d,1,r)) ).

Calogero-Moser spaces

Calogero-Moser spaces

c :Ref(W)/∼ −→ C

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

c_shy,s(x) −xis

Calogero-Moser spaces

c :Ref(W)/∼ −→ C

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

c_shy,s(x) −xis

Calogero-Moser spaces

c :Ref(W)/∼ −→ C

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

c_shy,s(x) −xis

Z_c =Z(H_c)

Z_c =Specmax(Zc) (Calogero-Moser space)

Calogero-Moser spaces

c :Ref(W)/∼ −→ C

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

c_shy,s(x) −xis

Z_c =Z(H_c)

Z_c =Specmax(Zc) (Calogero-Moser space) Example - Assume that c =0.

Calogero-Moser spaces

c :Ref(W)/∼ −→ C

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

c_shy,s(x) −xis

Z_c =Z(H_c)

Z_c =Specmax(Zc) (Calogero-Moser space) Example - Assume that c =0.

H₀ =C[V ×V^∗]⋊W

Calogero-Moser spaces

c :Ref(W)/∼ −→ C

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

c_shy,s(x) −xis

Z_c =Z(H_c)

Z_c =Specmax(Zc) (Calogero-Moser space)

Example - Assume that c =0.

H₀ =C[V ×V^∗]⋊W Z₀ =C[V ×V^∗]^W

Calogero-Moser spaces

c :Ref(W)/∼ −→ C

H_c =C[V]⊗CW ⊗C[V^∗] (as a vector space)

∀y ∈V, ∀x ∈V^∗, [y,x] = X

s∈Ref(W)

c_shy,s(x) −xis

Z_c =Z(H_c)

Z_c =Specmax(Zc) (Calogero-Moser space)

Example - Assume that c =0.

H₀ =C[V ×V^∗]⋊W Z₀ =C[V ×V^∗]^W Z₀ = (V ×V^∗)/W.

Calogero-Moser space (continued)

Calogero-Moser space (continued)

Theorem (Etingof-Ginzburg, 2002)

(a) Z_c is integral and integrally closed

Calogero-Moser space (continued)

Theorem (Etingof-Ginzburg, 2002)

(a) Z_c is integral and integrally closed (i.e. Z_c is irreducible and normal)

Calogero-Moser space (continued)

Theorem (Etingof-Ginzburg, 2002)

(a) Z_c is integral and integrally closed (i.e. Z_c is irreducible and normal)

(b) If Z_c is smooth, then

H_c-mod −→ Z_c-mod M 7−→ eM

is a Morita equivalence (here, e = _|W¹_|P

w∈W w).

Calogero-Moser space (continued)

Theorem (Etingof-Ginzburg, 2002)

(a) Z_c is integral and integrally closed (i.e. Z_c is irreducible and normal)

(b) If Z_c is smooth, then

H_c-mod −→ Z_c-mod

M 7−→ eM = (eH_c)⊗_Hc M is a Morita equivalence (here, e = _|W¹_|P

w∈W w).

Calogero-Moser space (continued)

Theorem (Etingof-Ginzburg, 2002)

(a) Z_c is integral and integrally closed (i.e. Z_c is irreducible and normal)

(b) If Z_c is smooth, then

H_c-mod −→ Z_c-mod

M 7−→ eM = (eH_c)⊗_Hc M is a Morita equivalence (here, e = _|W¹_|P

w∈W w).

(c) If Z_c is smooth, then





H²ⁱ⁺¹(Z_c) =0 H^2•(Z_c)|{z}≃

C-alg

GradFZ(CW)

Calogero-Moser space (continued)

Calogero-Moser space (continued)

H_c isZ-graded with

deg(V) = −1, deg(V^∗) =1 and deg(W) =0.

Calogero-Moser space (continued)

H_c isZ-graded with

deg(V) = −1, deg(V^∗) =1 and deg(W) =0.

⇒ Z_c is Z-graded (i.e. admits a C^×-action)

Calogero-Moser space (continued)

H_c isZ-graded with

deg(V) = −1, deg(V^∗) =1 and deg(W) =0.

⇒ Z_c is Z-graded (i.e. admits a C^×-action)

⇒ Z_c admits aC^×-action.

Calogero-Moser space (continued)

H_c isZ-graded with

deg(V) = −1, deg(V^∗) =1 and deg(W) =0.

⇒ Z_c is Z-graded (i.e. admits a C^×-action)

⇒ Z_c admits aC^×-action.

Theorem (Gordon, 2003)

IfZ_c issmooth, then

Z^C×

c

←→∼ Irr(W)

Equivariant cohomology

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

H^•_C×(pt)≃C[h], with deg(h) =2.

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

H^•_C×(pt)≃C[h], with deg(h) =2.

⇒ H^•_C×(X) is a C[h]-algebra.

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

H^•_C×(pt)≃C[h], with deg(h) =2.

⇒ H^•_C×(X) is a C[h]-algebra.

Theorem (folklore)

Assume thatH²ⁱ⁺¹(X) =0 for all i. Then:

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

H^•_C×(pt)≃C[h], with deg(h) =2.

⇒ H^•_C×(X) is a C[h]-algebra.

Theorem (folklore)

Assume thatH²ⁱ⁺¹(X) =0 for all i. Then:

(a) H²ⁱ_C×⁺¹(X) = 0 for alli.

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

H^•_C×(pt)≃C[h], with deg(h) =2.

⇒ H^•_C×(X) is a C[h]-algebra.

Theorem (folklore)

Assume thatH²ⁱ⁺¹(X) =0 for all i. Then:

(a) H²ⁱ_C×⁺¹(X) = 0 for alli.

(b) H^2•(X)≃C[h]/hhi ⊗H²_C^•×(X).

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

H^•_C×(pt)≃C[h], with deg(h) =2.

⇒ H^•_C×(X) is a C[h]-algebra.

Theorem (folklore)

Assume thatH²ⁱ⁺¹(X) =0 for all i. Then:

(a) H²ⁱ_C×⁺¹(X) = 0 for alli.

(b) H^2•(X)≃C[h]/hhi ⊗H²_C^•×(X).

(c) The canonical map i_X^∗ :H^2•_C×(X)−→H^2•_C×(X^C×)is injective.

Equivariant cohomology

If X is an algebraic variety endowed with a C^×-action, one can define its equivariant cohomologyH^•_C×(X).

H^•_C×(pt)≃C[h], with deg(h) =2.

⇒ H^•_C×(X) is a C[h]-algebra.

Theorem (folklore)

Assume thatH²ⁱ⁺¹(X) =0 for all i. Then:

(a) H²ⁱ_C×⁺¹(X) = 0 for alli.

(b) H^2•(X)≃C[h]/hhi ⊗H²_C^•×(X).

(c) The canonical map i_X^∗ :H^2•_C×(X)−→H^2•_C×(X^C×)is injective.

Remark - H²_C^•×(X^C×) =C[h]⊗H^2•(X).

Equivariant cohomology of smooth Z

Equivariant cohomology of smooth Z

Theorem (B.-Shan 2017)

IfZ_c issmooth, then (a) H²ⁱ_C×⁺¹(Z_c) =0.

(b) H²_C^•×(Z_c)|{z}≃

C-alg

ReesFZ(CW).

Equivariant cohomology of smooth Z

Theorem (B.-Shan 2017)

IfZ_c issmooth, then (a) H²ⁱ_C×⁺¹(Z_c) =0.

(b) H²_C^•×(Z_c)|{z}≃

C-alg

ReesFZ(CW).

Sketch of the proof. By folklore Theorem and Etingof-Ginzburg Theorem,

H²ⁱ_C×⁺¹(Z_c) =0

Equivariant cohomology of smooth Z

Theorem (B.-Shan 2017)

IfZ_c issmooth, then (a) H²ⁱ_C×⁺¹(Z_c) =0.

(b) H²_C^•×(Z_c)|{z}≃

C-alg

ReesFZ(CW).

Sketch of the proof. By folklore Theorem and Etingof-Ginzburg Theorem,

H²ⁱ_C×⁺¹(Z_c) =0

H^2•(Z_c)≃

C[h]/hhi ⊗H²_C^•×(Z_c)

Equivariant cohomology of smooth Z

Theorem (B.-Shan 2017)

IfZ_c issmooth, then (a) H²ⁱ_C×⁺¹(Z_c) =0.

(b) H²_C^•×(Z_c)|{z}≃

C-alg

ReesFZ(CW).

Sketch of the proof. By folklore Theorem and Etingof-Ginzburg Theorem,

H²ⁱ_C×⁺¹(Z_c) =0

H^2•(Z_c)≃

C[h]/hhi ⊗H²_C^•×(Z_c) GradFZ(CW)

Equivariant cohomology of smooth Z

Theorem (B.-Shan 2017)

IfZ_c issmooth, then (a) H²ⁱ_C×⁺¹(Z_c) =0.

(b) H²_C^•×(Z_c)|{z}≃

C-alg

ReesFZ(CW).

Sketch of the proof. By folklore Theorem and Etingof-Ginzburg Theorem,

H²ⁱ_C×⁺¹(Z_c) =0

H^2•(Z_c)≃

C[h]/hhi ⊗H²_C^•×(Z_c)

GradFZ(CW)≃C[h]/hhi ⊗ReesFZ(CW).

Sketch of the proof (continued)

H^2•(Z_c)≃

C[h]/hhi ⊗H^2•_C×(Z_c)

GradFZ(CW)≃C[h]/hhi ⊗ReesFZ(CW).

Sketch of the proof (continued)

H^2•(Z_c)≃

C[h]/hhi ⊗H^2•_C×(Z_c)

GradFZ(CW)≃C[h]/hhi ⊗ReesFZ(CW).

By folklore Theorem and Gordon Theorem, iZ^∗_c :H²C^•×(Z_c)֒−→H²C^•×(Z^C×

c )

Sketch of the proof (continued)

H^2•(Z_c)≃

C[h]/hhi ⊗H^2•_C×(Z_c)

GradFZ(CW)≃C[h]/hhi ⊗ReesFZ(CW).

By folklore Theorem and Gordon Theorem, iZ^∗_c :H²C^•×(Z_c)֒−→H²C^•×(Z^C×

c )≃C[h]⊗Z(CW).

Sketch of the proof (continued)

H^2•(Z_c)≃

C[h]/hhi ⊗H^2•_C×(Z_c)

GradFZ(CW)≃C[h]/hhi ⊗ReesFZ(CW).

By folklore Theorem and Gordon Theorem, iZ^∗_c :H²C^•×(Z_c)֒−→H²C^•×(Z^C×

c )≃C[h]⊗Z(CW).

An easy argument based on comparison of dimensions shows that it is sufficient to prove that

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

Sketch of the proof (continued)

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

Sketch of the proof (continued)

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

IfE is a graded CW-module, then H_c⊗^CW E is a graded projective H_c-module,

Sketch of the proof (continued)

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

IfE is a graded CW-module, then H_c⊗^CW E is a graded projective H_c-module, hence eH_c⊗CW E is a graded projective Z_c-module.

Sketch of the proof (continued)

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

IfE is a graded CW-module, then H_c⊗^CW E is a graded projective H_c-module, hence eH_c⊗CW E is a graded projective Z_c-module.

⇒equivariant Chern character ch(eH_c⊗CW E)∈H^²_C^•×(Z_c).

Sketch of the proof (continued)

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

IfE is a graded CW-module, then H_c⊗^CW E is a graded projective H_c-module, hence eH_c⊗CW E is a graded projective Z_c-module.

⇒equivariant Chern character ch(eH_c⊗CW E)∈H^²_C^•×(Z_c).

Theorem (Bellamy, 2009)

iZ^∗_c(ch(eH_c ⊗CW E)) = X

χ∈Irr(W)

q^bχhχ,C[V]^co(W)⊗Ei^gr_W hχ,C[V]^co(W)i^gr_W e_χ^W

where q =exp(h) and b_χ=valhχ,C[V]^co(W)i^gr_W.

Sketch of the proof (continued)

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

IfE is a graded CW-module, then H_c⊗^CW E is a graded projective H_c-module, hence eH_c⊗CW E is a graded projective Z_c-module.

⇒equivariant Chern character ch(eH_c⊗CW E)∈H^²_C^•×(Z_c).

Theorem (Bellamy, 2009)

iZ^∗_c(ch(eH_c ⊗CW E)) = X

χ∈Irr(W)

q^bχhχ,C[V]^co(W)⊗Ei^gr_W hχ,C[V]^co(W)i^gr_W e_χ^W

where q =exp(h) and b_χ=valhχ,C[V]^co(W)i^gr_W.

⇒(⋆) holds (!).

Sketch of the proof (continued)

(⋆) ReesFZ(CW)⊂Im(iZ^∗_c).

IfE is a graded CW-module, then H_c⊗^CW E is a graded projective H_c-module, hence eH_c⊗CW E is a graded projective Z_c-module.

⇒equivariant Chern character ch(eH_c⊗CW E)∈H^²_C^•×(Z_c).

Theorem (Bellamy, 2009)

iZ^∗_c(ch(eH_c ⊗CW E)) = X

χ∈Irr(W)

q^bχhχ,C[V]^co(W)⊗Ei^gr_W hχ,C[V]^co(W)i^gr_W e_χ^W

where q =exp(h) and b_χ=valhχ,C[V]^co(W)i^gr_W.

⇒(⋆) holds (!). QED

Back to example: W = S

Back to example: W = S

Theorem (B.-Maksimau, 2017)

Z₁(S_n)^µd =

Back to example: W = S

Theorem (B.-Maksimau, 2017)

Z₁(S_n)^µd = a

d-coresγ s.t. d|n−|γ|

Z_c

γ(G(d,1,rγ))

where r_γ = (n−|γ|)/d.

Back to example: W = S

Theorem (B.-Maksimau, 2017)

Z₁(S_n)^µd = a

d-coresγ s.t. d|n−|γ|

Z_c

γ(G(d,1,rγ))

where r_γ = (n−|γ|)/d. Fix such ad-coreγ.

Back to example: W = S

Theorem (B.-Maksimau, 2017)

Z₁(S_n)^µd = a

d-coresγ s.t. d|n−|γ|

Z_c

γ(G(d,1,rγ))

where r_γ = (n−|γ|)/d.

Fix such ad-coreγ. By functoriality of equivariant cohomology, we get two maps

H²_C^•×(Z₁(S_n)) ϕ^a_γ

//

 _

H²_C^•×(Z_c

γ(G(d,1,rγ)))

 _

H²_C^•×(Z₁(S_n)^C×) ϕ^b_γ

//H^2•_C×(Z_c

γ(G(d,1,rγ))^C×)

Back to example: W = S

(continued)

Back to example: W = S

(continued)

This leads to a commutative diagram

ReesFZ(CS_n) ϕ^a_γ

//

 _

ReesFZ(CG(d,1,rγ))

 _

C[h]⊗Z(CS_n) ϕ^b_γ

//C[h]⊗Z(CG(d,1,rγ))

Back to example: W = S

(continued)

This leads to a commutative diagram

ReesFZ(CS_n) ϕ^a_γ

//

 _

ReesFZ(CG(d,1,rγ))

 _

C[h]⊗Z(CS_n) ϕ^b_γ

//C[h]⊗Z(CG(d,1,rγ))

Using a theorem of Przezdziecki (2016), ϕ^b_γ =IdC[h]⊗ϕγ (!).

Back to example: W = S

(continued)

This leads to a commutative diagram

ReesFZ(CS_n) ϕ^a_γ

//

 _

ReesFZ(CG(d,1,rγ))

 _

C[h]⊗Z(CS_n) ϕ^b_γ

//C[h]⊗Z(CG(d,1,rγ))

Using a theorem of Przezdziecki (2016), ϕ^b_γ =IdC[h]⊗ϕγ (!). QED

Some comments

Some comments

(1) One can deduce from this work a proof of a conjecture of

Ginzburg-Kaledin on the cohomology ofsymplectic resolutions of (V ×V^∗)/W (B.-Shan, 2017).

Some comments

(1) One can deduce from this work a proof of a conjecture of

Ginzburg-Kaledin on the cohomology ofsymplectic resolutions of (V ×V^∗)/W (B.-Shan, 2017).

(2) With R. Rouquier (2017), we have general conjectures about

◮ equivariant cohomology of possibly singular Calogero-Moser spaces,

Some comments

(1) One can deduce from this work a proof of a conjecture of

Ginzburg-Kaledin on the cohomology ofsymplectic resolutions of (V ×V^∗)/W (B.-Shan, 2017).

(2) With R. Rouquier (2017), we have general conjectures about

◮ equivariant cohomology of possibly singular Calogero-Moser spaces,

◮ fixed points under the action of µ_d

Some comments

(1) One can deduce from this work a proof of a conjecture of

Ginzburg-Kaledin on the cohomology ofsymplectic resolutions of (V ×V^∗)/W (B.-Shan, 2017).

(2) With R. Rouquier (2017), we have general conjectures about

◮ equivariant cohomology of possibly singular Calogero-Moser spaces,

◮ fixed points under the action of µ_d

+ The combinatoric fits with observations made in the representation theory of finite reductive groups

Some comments

(1) One can deduce from this work a proof of a conjecture of

Ginzburg-Kaledin on the cohomology ofsymplectic resolutions of (V ×V^∗)/W (B.-Shan, 2017).

(2) With R. Rouquier (2017), we have general conjectures about

◮ equivariant cohomology of possibly singular Calogero-Moser spaces,

◮ fixed points under the action of µ_d

+ The combinatoric fits with observations made in the representation theory of finite reductive groups

◮ Lusztig families of unipotent characters

Some comments

(1) One can deduce from this work a proof of a conjecture of

Ginzburg-Kaledin on the cohomology ofsymplectic resolutions of (V ×V^∗)/W (B.-Shan, 2017).

(2) With R. Rouquier (2017), we have general conjectures about

◮ equivariant cohomology of possibly singular Calogero-Moser spaces,

◮ fixed points under the action of µ_d

+ The combinatoric fits with observations made in the representation theory of finite reductive groups

◮ Lusztig families of unipotent characters

◮ partition into blocks of unipotent characters (d-Harish-Chandra theory)

Some comments

(1) One can deduce from this work a proof of a conjecture of

Ginzburg-Kaledin on the cohomology ofsymplectic resolutions of (V ×V^∗)/W (B.-Shan, 2017).

(2) With R. Rouquier (2017), we have general conjectures about

◮ equivariant cohomology of possibly singular Calogero-Moser spaces,

◮ fixed points under the action of µ_d

+ The combinatoric fits with observations made in the representation theory of finite reductive groups

◮ Lusztig families of unipotent characters

◮ partition into blocks of unipotent characters (d-Harish-Chandra theory)

? Theory of spetses (Brou´e-Malle-Michel).