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

Binyong Sun Introduction to representation theory of classical Lie groups 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”.

Binyong Sun Introduction to representation theory of classical Lie groups Origin of group theory

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

x2 + bx + c = 0 has solutions √ −b ± b2 − 4c x = . 2 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.

Binyong Sun Introduction to representation theory of classical Lie groups Question

Can a given polynominal equation be solved by radicals?

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

Evariste´ Galois (1811õ1832)

Binyong Sun Introduction to representation theory of classical Lie groups Examples of groupsµ

(Z, +, 0) (Bijection(X , X ), ◦, 1)

(GLn(R), ·, 1)

Binyong Sun Introduction to representation theory of classical Lie groups Origin of Lie groups

Galois : Symetries of polynomial equations −→ Groups,

Sophus Lie : Symetries of differential equations −→ Lie groups.

Binyong Sun Introduction to representation theory of classical Lie groups Sophus Lie (1842õ1899)

Binyong Sun Introduction to representation theory of classical Lie groups Classical Lie groups

Most widely occurring Lie groups in mathematics and physics. Compact classical Lie groups

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

Binyong Sun Introduction to representation theory of classical Lie groups 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 ( " # " #) t 1p 0 1p 0 O(p, q) := g ∈ GLp+q(R) | g g = . 0 −1q 0 −1q

Binyong Sun Introduction to representation theory of classical Lie groups Finite dimensional representation theory:

1939

Binyong Sun Introduction to representation theory of classical Lie groups Hermann Weyl, 1885-1955

Binyong Sun Introduction to representation theory of classical Lie groups Two major achievements: Classical invariant theory; Classical branching law.

Binyong Sun Introduction to representation theory of classical Lie groups 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.

Binyong Sun Introduction to representation theory of classical Lie groups Standard representations:

n n 2n GLn(R) y C , GLn(C) y C , GLn(H) y C .

n n 2n O(n) y C , U(n) y C , Sp(n) y C .

Binyong Sun Introduction to representation theory of classical Lie groups 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.

Binyong Sun Introduction to representation theory of classical Lie groups 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.

Binyong Sun Introduction to representation theory of classical Lie groups Solution to duality problem:

Highest weight theory (Cartan).

Example Irr(U(1)) = Z, and more generally,

n Irr(U(n)) = {(a1 ≥ a2 ≥ · · · ≥ an) ∈ Z }.

Binyong Sun Introduction to representation theory of classical Lie groups Elie´ Cartan, 1869-1951

Binyong Sun Introduction to representation theory of classical Lie groups Example of spectral decomposition

2 1 Md U(1) y L (S ) = τk . k∈Z

Binyong Sun Introduction to representation theory of classical Lie groups 3. Classical invariant theory

n n O(n) y R ⇒ O(n) y C[R ].

Proposition

n O(n) C[R ] = C[q], where n X 2 q := xi . i=1

Binyong Sun Introduction to representation theory of classical Lie groups Problem n Decompose O(n) y C[R ]? Hidden symmetry

n O(n) × sl2(R) y C[R ],

" # 1 0 n Pn ∂ h := 7→ + i=1 xi , 0 −1 2 ∂xi " # 0 1 1 Pn 2 e := 7→ − i=1 xi , 0 0 2 " # 0 0 2 f := 7→ 1 Pn ∂ . 2 i=1 ∂x2 1 0 i

Binyong Sun Introduction to representation theory of classical Lie groups Harmonic polynomials

n n O(n) y H[R ] := {ϕ ∈ C[R ] | f · ϕ = 0} ∞ M k n = H [R ], k=0 where

k n n H [R ] := {ϕ ∈ H[R ] | ϕ is homogeneous of degree k}.

Binyong Sun Introduction to representation theory of classical Lie groups Theorem

n O(n) × sl2(R) y C[R ] L∞ k n n = k=0 H [R ] ⊗ L(k + 2 ).

Binyong Sun Introduction to representation theory of classical Lie groups More generally,

n×k n×k O(n) y R ⇒ O(n) y C[R ].

Hidden symmetry

n×k O(n) × sp2k (R) y C[R ],

Theorem (Classical invariant theory)

n×k M O(n) × sp2k (R) y C[R ] = τ ⊗ θ(τ). τ

Similar for other compact classical Lie groups.

Binyong Sun Introduction to representation theory of classical Lie groups 4. Infinite dimensional representations

Why? Harmonic analysis, quantum mechanics, number theory ··· .

Examples in Harmonic analysis. Fourier series: 2 1 U(1) y L (S ). Fourier transform:

n 2 n R y L (R ).

Automorphic forms:

2 GLn(R) y L (GLn(Z)\GLn(R)).

Binyong Sun Introduction to representation theory of classical Lie groups Founders:

Isra¨ılMoiseevich Gelfand, 1913-2009

Binyong Sun Introduction to representation theory of classical Lie groups Harish-Chandra, 1923-1983

Binyong Sun Introduction to representation theory of classical Lie groups Unitary representation: Hilbert space + unitary operators. Another example Stone’s Theorem

{Selfajoint operator on V } = {unitary rep. R y V } A 7→ (t 7→ eitA).

Binyong Sun Introduction to representation theory of classical Lie groups Example of irreducible rep.:

G = GLn(R), B := {upper triangular matrix} ⊂ G,

× χ : B → C a character. Then

∞ G y {f ∈ C (G) | f (bg) = χ(b) · f (g), b ∈ B, g ∈ G} is a representation which is irreducible for ”generic” χ.

Binyong Sun Introduction to representation theory of classical Lie groups 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.

Binyong Sun Introduction to representation theory of classical Lie groups Example of duality problem Langlands correspondence

× n Irr(GLn(C)) = {completely reducible rep. C y C }/ ∼

Binyong Sun Introduction to representation theory of classical Lie groups Robert Langlands

Binyong Sun Introduction to representation theory of classical Lie groups Examples of spectral decomposition Fourier series:

2 1 M n U(1) y L (S ) = d C · () . n∈Z Fourier transform: Z 2 n iξ·() L (R ) = C · e dξ, Rn

Binyong Sun Introduction to representation theory of classical Lie groups 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.

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

Binyong Sun Introduction to representation theory of classical Lie groups Roger Howe

Binyong Sun Introduction to representation theory of classical Lie groups Two fundamental conjectures Howe duality conjecture (Howe 1977) ( One-one correspondence, Multiplicity conservation.

Conservation relation conjecture of Kudla-Rallis (Kudla-Rallis 1994)

Binyong Sun Introduction to representation theory of classical Lie groups Local symmetry:

(p+q)×k O(p, q) × sp2k (R) y C[R ].

Global symmetry:

(p+q)×k O(p, q) × Spf2k (R) y S(R ).

Binyong Sun Introduction to representation theory of classical Lie groups Write

(p+q)×k ωk : = S(R )

Ω : = {π ∈ Irr(O(p, q)) | HomO(p,q)(ωk , π) 6= 0}, 0 0 0 Ω : = {π ∈ Irr(Spf2k ( )) | Hom (ωk , π ) 6= 0}. R Spf2k (R)

Binyong Sun Introduction to representation theory of classical Lie groups Theorem [Howe, JAMS 1989] One-one correspondenceµThe relation

0 Hom (ωk , π⊗π ) 6= 0 O(p,q)×Spf2k (R) b

yields a one-one correspondence

0 Irr(O(p, q)) ⊃ Ω ↔ Ω ⊂ Irr(Spf2k (R)).

Multiplicity preservationµfor all (π, π0) ∈ Ω × Ω0,

0 Hom (ωk , π⊗π ) = 1. O(p,q)×Spf2k (R) b

Binyong Sun Introduction to representation theory of classical Lie groups 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).

Binyong Sun Introduction to representation theory of classical Lie groups Summary of theta correspondence Transfer representations of one classical group to another classical group.

Binyong Sun Introduction to representation theory of classical Lie groups 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)

Binyong Sun Introduction to representation theory of classical Lie groups Problem. Given π ∈ Irr(O(p, q)),

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

Binyong Sun Introduction to representation theory of classical Lie groups 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.

Binyong Sun Introduction to representation theory of classical Lie groups First occurrence index:

n(π) := min{k | HomO(p,q)(ωk , π) 6= 0}.

Example.

n(1) = 0, n(det) = p + q (Weyl, Rallis, Przebinda).

Binyong Sun Introduction to representation theory of classical Lie groups 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.

Binyong Sun Introduction to representation theory of classical Lie 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).

Binyong Sun Introduction to representation theory of classical Lie groups 6. Classical branching law

Two methods of constructing representationsµ Induction, restriction.

Restriction ↔ Symmetry breaking.

Binyong Sun Introduction to representation theory of classical Lie groups Theorem (Classical branching law)

Let τµ ∈ Irr(U(n))§then M (τµ)|U(n−1) = τν. ν4µ Similar result holds for orthogonal groups.

Binyong Sun Introduction to representation theory of classical Lie groups Proof • Classical invariant theory. Application • Basis of irreducible representation.

Binyong Sun Introduction to representation theory of classical Lie groups 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)

Binyong Sun Introduction to representation theory of classical Lie groups 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).

Binyong Sun Introduction to representation theory of classical Lie groups 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)

Binyong Sun Introduction to representation theory of classical Lie groups 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)).

Binyong Sun Introduction to representation theory of classical Lie groups 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)

Binyong Sun Introduction to representation theory of classical Lie groups 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.

Binyong Sun Introduction to representation theory of classical Lie groups L-function: generalization of Riemann zeta function Langlands program:

Binyong Sun Introduction to representation theory of classical Lie groups Thank you!

Binyong Sun Introduction to representation theory of classical Lie groups