Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and...

Atlas of Lie Groups and Representations

Jeffrey AdamsSun Yat-sen University, Guanhzhou

August 4, 2018

Slides available at: www.liegroups.org

Atlas Project

Jeffrey Adams

Dan Barbasch

Birne Binegar

Fokko du Cloux

Marc van Leeuwen

Marc Noel

Annegret Paul

Susana Salamanca

Siddhartha Sahi

John Stembridge

Peter Trapa

David Vogan

Overview

Atlas software:

Computations in Lie theory

Computing unitary representations, the unitary dual, many otherthings

Software is freely available www.liegroups.org

See www.liegroups.org/help

1) Learn by examples.

2) You only really understand something if you can program it on acomputer.

Overview

Atlas software:


Computing unitary representations, the unitary dual,

many otherthings





Overview

Atlas software:


Computing unitary representations, the unitary dual, many otherthings





Before we begin. . .

Note: Everything is in the context of G (C) is a connected complexreductive group,

G (R) a real form of G (C)

K (R) ⊂ G (R) maximal compact, K (C) ⊂ G (C)

“forget about K (R)”:everything in terms of G (C),K (C):

Unless otherwise noted: (almost) everything in sight is complexand complex algebraic





“forget about K (R)”:

everything in terms of G (C),K (C):






“forget about K (R)”:everything in terms of G (C),K (C):


Structure Theory

1. Root data and reductive groups

1.1 Root Data1.2 roots, coroots, simple roots, highest root,fundamental

weights. . .1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group

Structure Theory


1.1 Root Data

1.2 roots, coroots, simple roots, highest root,fundamentalweights. . .

1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group

Structure Theory



weights. . .

1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group

Structure Theory



weights. . .1.3 Weyl group

1.4 Complex connected reductive groups1.5 Radical, center, derived group

Structure Theory



weights. . .1.3 Weyl group1.4 Complex connected reductive groups

1.5 Radical, center, derived group

Structure Theory



weights. . .1.3 Weyl group1.4 Complex connected reductive groups1.5 Radical, center, derived group

Real Groups

2 Involutions of reductive groups, real groups

2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms

Example:G = SL(n,C)

Out(G ) = Z2 = {1, γ}

Inner class of 1: {SU(p, q) | p + q = n}

Inner class of γ (γ(g) =t g−1):SL(n,R), also SL(n/2,H)

Real Groups


2.1 Cartan involution

2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms

Example:G = SL(n,C)

Out(G ) = Z2 = {1, γ}



Real Groups


2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 1

2.3 Pinnings, splitting of the exact sequence2.4 Inner classes of real forms

Example:G = SL(n,C)

Out(G ) = Z2 = {1, γ}



Real Groups


2.1 Cartan involution2.2 1→ Int(G )→ Aut(G )→ Out(G )→ 12.3 Pinnings, splitting of the exact sequence

2.4 Inner classes of real forms

Example:G = SL(n,C)

Out(G ) = Z2 = {1, γ}



Real Groups



Example:G = SL(n,C)

Out(G ) = Z2 = {1, γ}



Real Groups



Example:G = SL(n,C)

Out(G ) = Z2 = {1, γ}


Inner class of γ (γ(g) =t g−1):SL(n,R),

also SL(n/2,H)

Real Groups



Example:G = SL(n,C)

Out(G ) = Z2 = {1, γ}



K

3 Maximal compact subgroup K (R)

3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0

3.4 K orbits on G/B (B=Borel subgroup)

Example: G = GL(n,C)

θ(g) = Jp,qgJ−1p,q ,

Jp,q = diag(Ip,−Iq),K = GL(p,C)× GL(q,C),

K (R) = U(p)× U(q), G (R) = U(p, q)

θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)

Theorem: G/B = G (C)/B(C) is a projective variety, with finitelymany K = K (C)-orbits

3.5 Real Cartan subgroups, (relative) Weyl groups

K


3.1 Maximal compact subgroup

3.2 K (C) = G θ, K (R) compact3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0





K (R) = U(p)× U(q), G (R) = U(p, q)




K


3.1 Maximal compact subgroup3.2 K (C) = G θ, K (R) compact

3.3 Component groups: K/K 0 = K (R)/K (R)0 = G (R)/G (R)0





K (R) = U(p)× U(q), G (R) = U(p, q)




K







K (R) = U(p)× U(q), G (R) = U(p, q)




K






Jp,q = diag(Ip,−Iq),

K = GL(p,C)× GL(q,C),

K (R) = U(p)× U(q), G (R) = U(p, q)




K







K (R) = U(p)× U(q), G (R) = U(p, q)




K







K (R) = U(p)× U(q)

, G (R) = U(p, q)




K







K (R) = U(p)× U(q), G (R) = U(p, q)




K







K (R) = U(p)× U(q), G (R) = U(p, q)

θ(g) =t g−1,

K (C) = O(n,C), K (R) = O(n), G (R) = GL(n,R)



K







K (R) = U(p)× U(q), G (R) = U(p, q)

θ(g) =t g−1, K (C) = O(n,C),

K (R) = O(n), G (R) = GL(n,R)



K







K (R) = U(p)× U(q), G (R) = U(p, q)

θ(g) =t g−1, K (C) = O(n,C), K (R) = O(n),

G (R) = GL(n,R)



K







K (R) = U(p)× U(q), G (R) = U(p, q)




Example: SO(4,4)

Langlands classification

4 Langlands Classification

4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with

infinitesimal character γ)Harish-Chandra: finite set

4.6 Cuspidal data: π = IndG

P (πM)

4.7 theta-stable data: π = RGq (πL)



4.1 Parameters for representations

4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with



P (πM)




4.1 Parameters for representations4.2 (g,K )-modules

, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with



P (πM)




4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)

4.3 Admissible representations4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with



P (πM)




4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations

4.4 Infinitesimal character4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation with



P (πM)




4.1 Parameters for representations4.2 (g,K )-modules, Note: G = G (C),K = K (C)4.3 Admissible representations4.4 Infinitesimal character

4.5 Mγ = {γ1, . . . , γn} (parameters for irreducible representation withinfinitesimal character γ)Harish-Chandra: finite set


P (πM)





infinitesimal character γ)

Harish-Chandra: finite set


P (πM)







P (πM)


Kazhdan-Lusztig-Vogan Polynomials

5 Kazhdan-Lusztig-Vogan Polynomials

5.1 Composition series: I (Γ) =∑

mΞ,ΓJ(Ξ)5.2 Character formula: J(Γ) =

∑MΞ,ΓI (Ξ)

5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)




mΞ,ΓJ(Ξ)

5.2 Character formula: J(Γ) =∑

MΞ,ΓI (Ξ)5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)





∑MΞ,ΓI (Ξ)






∑MΞ,ΓI (Ξ)

5.3 PΞ,Γ ∈ Z[q]

5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)





∑MΞ,ΓI (Ξ)

5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]

5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)





∑MΞ,ΓI (Ξ)

5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)

5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)





∑MΞ,ΓI (Ξ)

5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)

5.7 Global character (function on regular semisimple set)5.8 (Twisted KLV polynomials)





∑MΞ,ΓI (Ξ)

5.3 PΞ,Γ ∈ Z[q]5.4 QΞ,Γ ∈ Z[q]5.5 mΞ,Γ = QΞ,Γ(q = 1)5.6 MΞ,Γ = ±PΞ,Γ(q = 1)5.7 Global character (function on regular semisimple set)

5.8 (Twisted KLV polynomials)





∑MΞ,ΓI (Ξ)


Unitary representations

6 Hermitian forms and unitary representations

6.1 Hermitian representations (preserving a Hermitian form)6.2 Unitary representations (preserving a positive definite Hermitian

form)

6.3 Gd ⊂ Gtemp ⊂ Gherm ⊂ Gunitary ⊂ Gadm

(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible

quotient of I (Γ)



6.1 Hermitian representations (preserving a Hermitian form)

6.2 Unitary representations (preserving a positive definite Hermitianform)



quotient of I (Γ)




form)



quotient of I (Γ)




form)


(discrete series/tempered/Hermitian/unitary/admissible)

6.4 Γ ∈ Mγ → I (Γ) (standard module) J(Γ) (unique irreducible

quotient of I (Γ)




form)


(discrete series/tempered/Hermitian/unitary/admissible)6.4 Γ ∈ Mγ → I (Γ) (standard module)

J(Γ) (unique irreducible

quotient of I (Γ)




form)



quotient of I (Γ)

Branching to K

7 Branching to K

7.1 Parametrization of K (K (R),K (C) are possibly disconnected)7.2 Compute standard module I (Γ)|K =

∑ni=1 aiµi (up to some

height bound)7.3 Compute irreducible module J(Γ)|K =


height bound)

Branching to K

7 Branching to K

7.1 Parametrization of K (K (R),K (C) are possibly disconnected)

7.2 Compute standard module I (Γ)|K =∑n

i=1 aiµi (up to someheight bound)

7.3 Compute irreducible module J(Γ)|K =∑n


Branching to K

7 Branching to K



height bound)

7.3 Compute irreducible module J(Γ)|K =∑n


Branching to K

7 Branching to K



height bound)7.3 Compute irreducible module J(Γ)|K =


height bound)

Parabolic Induction

8 Real parabolic induction

8.1 real parabolic subgroups8.2 IndG

P (πM) =∑

aipii (in the Grothendieck group)8.3 Reducibilty of IndG

P (πM , ν) (in terms of ν)

9 Cohomological induction

9.1 theta-stable parabolic subgroups9.2 RG

q (πL) =∑

aipii (in the Grothendieck group)

9.3 More precisely:∑

i (−1)iRG ,iq (πL)

9.4 Using: coherent continuation

Parabolic Induction


8.1 real parabolic subgroups

8.2 IndGP (πM) =

∑aipii (in the Grothendieck group)

8.3 Reducibilty of IndGP (πM , ν) (in terms of ν)



q (πL) =∑





Parabolic Induction



P (πM) =∑


8.3 Reducibilty of IndGP (πM , ν) (in terms of ν)



q (πL) =∑





Parabolic Induction



P (πM) =∑





q (πL) =∑





Parabolic Induction



P (πM) =∑




9.1 theta-stable parabolic subgroups

9.2 RGq (πL) =

∑aipii (in the Grothendieck group)




Parabolic Induction



P (πM) =∑





q (πL) =∑





Parabolic Induction



P (πM) =∑





q (πL) =∑


9.3 More precisely:

∑i (−1)iRG ,i

q (πL)9.4 Using: coherent continuation

Parabolic Induction



P (πM) =∑





q (πL) =∑





Example G = Sp(12,R), M = GL(5,R)× SL(2,R)

atlas> G:=Sp(12,R)

Value: connected split real group with Lie algebra ’sp(12,R)’

atlas> set real_parabolics=all_real_parabolics (G)

Variable real_parabolics: [KGPElt]

atlas> #real_parabolics

Value: 64

atlas> void:for P@i in real_parabolics do if

ss_rank (Levi(P))=5 then prints(i, " ", Levi(P)) fi od

31 sl(6,R).gl(1,R)

47 sl(5,R).sl(2,R).gl(1,R)

55 sl(4,R).sp(4,R).gl(1,R)

59 sl(3,R).sp(6,R).gl(1,R)

61 sl(2,R).sp(8,R).gl(1,R)

62 sp(10,R).gl(1,R)

atlas> set P=real_parabolics[47]

Variable P: KGPElt

atlas> real_induce_irreducible(trivial(Levi(P)),G)

Value:

1*parameter(x=4898,lambda=[6,5,4,3,2,1]/1,nu=[2,2,1,1,1,0]/1)



Nilpotent Orbits

10. Nilpotent orbits

10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of

centralizer, component group10.5 coming soon: real nilpotent: identity component of reductive

part of centralizer, component group

Nilpotent Orbits


10.1 parametrize Complex nilpotent orbits

10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of



Nilpotent Orbits


10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits

10.3 Note: These parametrizations are case-free, in atlas language10.4 complex nilpotent: identity component of reductive part of



Nilpotent Orbits


10.1 parametrize Complex nilpotent orbits10.2 parametrize real nilpotent orbits10.3 Note: These parametrizations are case-free, in atlas language

10.4 complex nilpotent: identity component of reductive part ofcentralizer, component group

10.5 coming soon: real nilpotent: identity component of reductivepart of centralizer, component group

Nilpotent Orbits



centralizer, component group

10.5 coming soon: real nilpotent: identity component of reductivepart of centralizer, component group

Nilpotent Orbits





Advanced topics

11. Advanced topics (selected)

11.1 Weyl character formula11.2 Frobenius-Schur and quaternionic indicators for finite

dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W

Advanced topics


11.1 Weyl character formula

11.2 Frobenius-Schur and quaternionic indicators for finitedimensional representation

11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W

Advanced topics



dimensional representation

11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W

Advanced topics



dimensional representation11.3 Conjugacy classes in W (nice representatives)

11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W

Advanced topics



dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W

11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W

Advanced topics



dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W

11.6 Cell representations11.7 Special representations of W

Advanced topics



dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations

11.7 Special representations of W

Advanced topics



dimensional representation11.3 Conjugacy classes in W (nice representatives)11.4 Character table of W11.5 Coherent continuation representation of W11.6 Cell representations11.7 Special representations of W

11. Advanced topics (continued)

11.7 AVann(π) “complex associated variety”, closure of a singlecomplex orbit (π irreducible)

11.8 Primitive ideals11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbits

of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations

12. p-adic groups. . . (Sorry, just kidding)

Thank You



11.8 Primitive ideals

11.9 AV (π) “real associated variety”: {O1, . . . ,Oi} nilpotent orbitsof G (R) on gR or K orbits on s

11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations


Thank You




of G (R) on gR or K orbits on s

11.10 “Weak” Arthur packets of special unipotent representations11.11 “Honest” Arthur packets of special unipotent representations


Thank You




of G (R) on gR or K orbits on s11.10 “Weak” Arthur packets of special unipotent representations

11.11 “Honest” Arthur packets of special unipotent representations


Thank You






Thank You





12. p-adic groups. . .

(Sorry, just kidding)

Thank You






Thank You

Atlas of Lie Groups and Representations · 2018. 10. 20. · Atlas of Lie Groups and...

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