introduction to representation theory of classical lie groups · 1.classical lie groups...

Introduction to representation theory of

classical Lie groups

Binyong Sun

Academy of Mathematics and Systems Science, Chinese Academy of Sciences

The Tsung-Dao Lee Institute

2019.7.19

Binyong Sun Introduction to representation theory of classical Lie groups

Contents

1. Classical Lie groups

2. Representations of compact Lie groups

3. Classical invariant theory

4. Infinite dimensional representations

5. Theta correspondences

6. Classical branching law

7. More questions

1.Classical Lie groups

Symmetry: Groups and their representations.

P. W. Anderson (Nobel prize winner)µ

“It is only slightly overstating the case that

physics is the study of symmetry”.

Origin of group theory

Babylonian era (2000 B.C.): the quadratic equation

x2 + bx + c = 0

has solutions

x =−b ±

√b2 − 4c

16 century: The cubic and quartic equations can be solved by

radicals.

AbelõRuffini Theorem(1799!1824): Quintic or higher degree

equations can not be solved by radical in general.

Question

Can a given polynominal equation be solved by radicals?

Galois (1830s): establish group theory; answer this question.

Evariste Galois (1811õ1832)

Examples of groupsµ

(Z,+, 0)

(Bijection(X ,X ), , 1)

(GLn(R), ·, 1)

Origin of Lie groups

Galois :

Symetries of polynomial equations −→ Groups,

Sophus Lie :

Symetries of differential equations −→ Lie groups.

Sophus Lie (1842õ1899)

Classical Lie groups

Most widely occurring Lie groups in mathematics and physics.

Compact classical Lie groups

O(n) = g ∈ GLn(R) | gg t = 1n,U(n) = g ∈ GLn(C) | gg t = 1n,Sp(n) = g ∈ GLn(H) | gg t = 1n.

Real classical groups

General linear group:

GLn(R), GLn(C), GLn(H),

Real orthogonal groups and so on:

O(p, q), Sp2n(R), U(p, q), O∗(2n), Sp(p, q).

Complex orthogonal groups and complex symplectic groups:

On(C), Sp2n(C).

Example

O(p, q) :=

g ∈ GLp+q(R) | g t

0 −1q

Finite dimensional representation theory:

Hermann Weyl, 1885-1955

Two major achievements:

Classical invariant theory;

Classical branching law.

2.Representations of compact Lie groups

Definition

Let G be a group. A representation of G is a complex vector space

V , together with a linear action

G × V → V , (g , v) 7→ g .v .

Notation: G y V .

Continuity condition.

Linear action:

g .(au + bv) = a(g .u) + b(g .v),

(gh).u = g .(h.u),

1.u = u.

Standard representations:

GLn(R) y Cn, GLn(C) y Cn, GLn(H) y C2n.

O(n) y Cn, U(n) y Cn, Sp(n) y C2n.

Analogy

Positive integers : Representations

prime numbers : Irreducible representations

Definition

A representation is said to be irreducible if it is nonzero, and has

no proper nonzero subrepresentation.

+ topological condition.

G : compact Lie group.

Two basic problemsµ

Duality problem: Calculate

Irr(G ) := Irreducible representation of G/ ∼ .

Spectral decomposition : Given G y V , write V as a sum of

irreducible representations.

Solution to duality problem:

Highest weight theory (Cartan).

Example

Irr(U(1)) = Z,

and more generally,

Irr(U(n)) = (a1 ≥ a2 ≥ · · · ≥ an) ∈ Zn.

Elie Cartan, 1869-1951

Example of spectral decomposition

U(1) y L2(S1) =⊕

k∈Zτk .

3. Classical invariant theory

O(n) y Rn ⇒ O(n) y C[Rn].

Proposition

C[Rn]O(n) = C[q],

q :=n∑

Problem

Decompose O(n) y C[Rn]?

Hidden symmetry

O(n)× sl2(R) y C[Rn],

0 −1

]7→ n

2 +∑n

i=1 xi∂∂xi,

]7→ −1

∑ni=1 x

]7→ 1

∑ni=1

∂x2i.

Harmonic polynomials

O(n) y H[Rn] := ϕ ∈ C[Rn] | f · ϕ = 0

=∞⊕k=0

Hk [Rn],

Hk [Rn] := ϕ ∈ H[Rn] | ϕ is homogeneous of degree k.

Theorem

O(n)× sl2(R) y C[Rn]

=⊕∞

k=0Hk [Rn]⊗ L(k + n2 ).

More generally,

O(n) y Rn×k ⇒ O(n) y C[Rn×k ].

Hidden symmetry

O(n)× sp2k(R) y C[Rn×k ],

Theorem (Classical invariant theory)

O(n)× sp2k(R) y C[Rn×k ] =⊕τ

τ ⊗ θ(τ).

Similar for other compact classical Lie groups.

4. Infinite dimensional representations

Why? Harmonic analysis, quantum mechanics, number theory

· · · .

Examples in Harmonic analysis.

Fourier series:

U(1) y L2(S1).

Fourier transform:

Rn y L2(Rn).

Automorphic forms:

GLn(R) y L2(GLn(Z)\GLn(R)).

Founders:

Israıl Moiseevich Gelfand, 1913-2009

Harish-Chandra, 1923-1983

Unitary representation: Hilbert space + unitary operators.

Another example

Stone’s Theorem

Selfajoint operator on V = unitary rep. R y V A 7→ (t 7→ e itA).

Example of irreducible rep.:

G = GLn(R),

B := upper triangular matrix ⊂ G ,

χ : B → C× a character.

G y f ∈ C∞(G ) | f (bg) = χ(b) · f (g), b ∈ B, g ∈ G

is a representation which is irreducible for ”generic” χ.

G : Lie group.

Two basic problemsµ

Duality problem: Calculate

Irr(G ) := “Irreducible rep.” of G/ ∼⊃ Irru(G ) := Irreducible unitary rep. of G/ ∼ .

Spectral decomposition : Given G y V , write V as a sum of

irreducible representations.

Example of duality problem

Langlands correspondence

Irr(GLn(C)) = completely reducible rep. C× y Cn/ ∼

Robert Langlands

Examples of spectral decomposition

Fourier series:

U(1) y L2(S1) =⊕

n∈ZC · ( )n.

Fourier transform:

L2(Rn) =

C · e iξ·( ) dξ,

5. Theta correspondence

Classical invariant theory ←→ Theta correspondence

compact group ←→ real or p-adic group,

polynomial function ←→ generalized function,

finite dim. rep. ←→ infinite dim. rep.,

local symmetry ←→ global symmetry,

H. Weyl ←→ R. Howe.

p-adic fields: Q completion−−−−−−−−→ R, Q2, Q3, Q5, Q7, Q11, · · · .

Roger Howe

Two fundamental conjectures

Howe duality conjecture (Howe 1977)One-one correspondence,

Multiplicity conservation.

Conservation relation conjecture of Kudla-Rallis (Kudla-Rallis

Local symmetry:

O(p, q)× sp2k(R) y C[R(p+q)×k ].

Global symmetry:

O(p, q)× Sp2k(R) y S(R(p+q)×k).

ωk : = S(R(p+q)×k)

Ω : = π ∈ Irr(O(p, q)) | HomO(p,q)(ωk , π) 6= 0,

Ω′ : = π′ ∈ Irr(Sp2k(R)) | HomSp2k (R)

(ωk , π′) 6= 0.

Theorem [Howe, JAMS 1989]

One-one correspondenceµThe relation

HomO(p,q)×Sp2k (R)

(ωk , π⊗π′) 6= 0

yields a one-one correspondence

Irr(O(p, q)) ⊃ Ω↔ Ω′ ⊂ Irr(Sp2k(R)).

Multiplicity preservationµfor all (π, π′) ∈ Ω× Ω′,

HomO(p,q)×Sp2k (R)

(ωk , π⊗π′) = 1.

Theorem (Howe duality conjecture)

The same holds for all real or p-adicd classical groups.

The real case: Howe, JAMS 1989.

p-adic case§p 6= 2: Waldspurger, Proceeding for 60’s birthday

of Piatetski-Shapiro, 1990.

Orthogonal, symplectic, unitary groups (Multiplicity

perservation)µLi-Sun-Tian, Invent. Math. (2011),

Orthogonal, symplectic, unitary groups: Gan-Takeda, JAMS

(2015).

The last caseµGan-Sun, Proceeding for 70’s birthday of Howe

(2017).

Summary of theta correspondence

Transfer representations of one classical group to another

classical group.

Applications of theta correspondence

Constructions of unitary representations:

• Jian-Shu Li§Invent. Math. (1989)

• Ma-Sun-Zhu§preprint (2017)

Constructions of automorphic representations:

• Howe, Proc. Sympos. Pure Math. (1979)

• Harris-Kudla-Sweet, JAMS (1996)

L-functions:

• Kudla-Rallis, Ann. of Math. (1994)

• Gan-Qiu-Takeda, Invent. Math. (2014)

Problem. Given π ∈ Irr(O(p, q)),

HomO(p,q)(ωk , π) 6= 0 ? (ωk := S(R(p+q)×k).

Kulda persistence principle:

HomO(p,q)(ωk , π) 6= 0 ⇒ HomO(p,q)(ωk+1, π) 6= 0.

Howe’s stable range:

k ≥ p + q ⇒ HomO(p,q)(ωk , π) 6= 0.

First occurrence index:

n(π) := mink | HomO(p,q)(ωk , π) 6= 0.

Example.

n(1) = 0,

n(det) = p + q (Weyl, Rallis, Przebinda).

Theorem (Kudla-Rallis’s conservation relation conjecture),

Sun-Zhu, JAMS (2015)

n(π) + n(π ⊗ det) = p + q.

The same holds for all real or p-adic classical groups.

Applications of the conservations relations:

The final proof of Howe duality conjecture

• Gan-Sun, proceeding for 70’s birthday of Howe

(2017)

Explicit calculation of theta correspondence

• Atobe-Gan, Invent. Math. (2017)

Zeros and poles and L-functions

• Yamana, Invent. Math. (2014)

Local Landlands correspondence

• Gan-Ichino, Ann. of Math. (2018).

6. Classical branching law

Two methods of constructing representationsµ

Induction, restriction.

Restriction↔ Symmetry breaking.

Theorem (Classical branching law)

Let τµ ∈ Irr(U(n))§then

(τµ)|U(n−1) =⊕ν4µ

τν .

Similar result holds for orthogonal groups.

• Classical invariant theory.

Application

• Basis of irreducible representation.

Uniqueness of branching. For all

τµ ∈ Irr(U(n))§τν ∈ Irr(U(n − 1)),

dim HomU(n−1)(τµ, τν) ≤ 1.

Multiplicity one theorem

Similar results holds for all real or p-adic classical groups.

Conjectured: Bernstein-Rallis, 1980’s

p-adic case: Aizenbud-Gourevitch-Rallis-Schiffmann, Ann. of

Math. (2010)

real case: Sun-Zhu, Ann. of Math. (2012)

Example (Waldspurger formula and Gross-Zagier formula).

Infinite dimensional representation

GL2(R) y π.

One dimensional representation

GL1(R) y χ.

uniqueness of branching

dim HomGL1(R)(π, χ) = 1.

This is the Rankin-Selberg theory for GL2(R)×GL1(R).

Jacobi groupµ

GLn(R) n H2n+1(R), Sp2n(R) n H2n+1(R).

Multiplicity one theorem for Jacobi groups)

Similar result holds for real or p-adic Jacobi groups.

Conjectured: Prasad, 1990’s

p-adic case: Sun, Amer. J. of Math. (2012)

real case: Sun-Zhu, Ann. of Math. (2012)

Example (Tate thesis).

Irreducible representation:

GL1(R) n H3(R) y S(R).

One dimensional representation

GL1(R) y χ.

Uniqueness of homogeneous generalized functions:

dim HomGL1(R)(S(R), χ) = 1.

This is theta correspondence for (GL1(R),GL1(R)).

Applications of the multiplicity one theorem

Local Gan-Gross-Prasad conjecture

• Hongyu He, Invent. Math. (2017)

Global Gan-Gross-Prasad conjecture

• Wei Zhang, Ann. of Math. (2014)

Proof of Kazhdan-Mazur’s nonvanishing hypothesis

• Sun, JAMS (2017)

8. More questions

Determine theta correspondence.

Local Gan-Gross-Prasad conjecture

Global Gan-Gross-Prasad cojecture

Unitary dual of classical Lie groups

Algebraic automorphic representations and arithmetic of

L-functions.

L-function: generalization of Riemann zeta function

Langlands program:

Thank you!

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