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We say that “σ occurs in V ” if m(σ, V ) = 0. Denote the set of all K-types occurring in V by (K,V ) = σ (K) : m(σ, V )6 = 0 . A finitely generated (g, K)-module V is called admissibleR if{ every∈ Rσ (K) has finite6 } multiplicity in V . It is well-known that every irreducible (g, K)-module∈ R is admissible. A real is a pair (G, G′) of closed reductive subgroups of Sp = Sp2N (R) (for some N) such that they are mutual centralizers of each other. For a subgroup E Sp, let E denote its preimage in the metaplectic cover (the unique non- ⊆ trivial two-fold central extension) Sp of Sp. Indeed we have short exact sequence: e 1 fµ Sp Sp 1, → 2 → → → where µ = Ker(Sp Sp) is the finitef group of order 2. Take the Segal-Shale-Weil 2 → oscillator representation (c.f. [Sha62, Wei64, LV80]) ω of Sp (associated to the character f of R that sends t to exp(2π√ 1t)). Let sp denote the complexified Lie algebra of Sp, f and take U = U(N) as a maximal− compact subgroup of Sp. Fock model realizes the (sp, U)-module of ω on the space = Poly(CN ) of complex polynomials on CN . Assume that G and G′ are embeddedF in Sp in such a way that K = U G and e ∩ K′ = U G′ are maximal compact subgroups of G and G′ respectively. Hence K and ∩ K′ are maximal compact subgroups of G and G′ respectively. Let g and g′ be the e complexified Lie algebras of G and G′ respectively. f e f Lemma 1.1 (Algebraic version of local theta correspondence over R [How89b, Theorem 2.1]). For a real reductive dual pair (G, G′), if π is an irreducible (g, K)-module, then e Ker(T ) π Θ(π), F ≃ ⊗ ∈ F T Hom\g,Ke ( ,π) where Θ(π) is a finitely generated admissible (g′, K′)-module. Moreover, if Θ(π) is non- g′ ′ zero, then it has a unique irreducible ( , K )-quotientf θ(π) called the “theta lift” of π. The intersection in the quotientf may be the whole . This case happens • F ⇔ Hom e ( ,π) = 0 Θ(π) = 0. In this case, let the theta lift θ(π) = 0. g,K F ⇔ When Θ(π) = 0, the “uniqueness” of θ(π) means that Θ(π) has a unique maximal • 6 proper sub-(g′, K′)-module, and it is the kernel of Θ(π) θ(π). → Recall that the oscillatorf representation ω is the direct sum of two irreducible unitary representations of Sp, and we have a (sp, U)-module decomposition = , F F0 ⊕ F1 where i is the linear span of all homogeneous polynomials of degree i (mod 2) in F f e ≡ = Poly(CN ), for i 0, 1 . Then as (g′, K′)-modules Θ(π)=Θ (π) Θ (π), with F ∈ { } 0 ⊕ 1 e i Ker(T ) π Θi(π). F  ≃ ⊗ ∈ F T Hom\g,Ke ( i,π)

Proposition 1.2. Let π be an irreducible (g, K)-module. Then Θi(π) = 0 for at least one i 0, 1 . In other words, Hom e ( i,π) = 0 for this i. ∈ { } g,K F e PARITY PRESERVATION OF K-TYPES UNDER THETA CORRESPONDENCE 3

Proof. Otherwise, Θ(π)=Θ0(π) Θ1(π) with both Θi(π) non-zero finitely generated ′ ′ ⊕ admissible (g , K )-modules. By Zorn’s Lemma, Θi(π) contains a maximal proper sub- ′ ′ (g , K )-module Πi. Then both Θ(π)/(Π Θ (π)) and Θ(π)/(Θ (π) Π ) are non-zero f 0 ⊕ 1 0 ⊕ 1 g′ ′  irreduciblef ( , K )-quotients, in contradiction with the uniqueness in Lemma 1.1. Corollary 1.3.f (K, ) (K, )=Ø. R F0 ∩ R F1 Proof. Let M be thee centralizere of K in Sp, then (K, M) is also a reductive dual pair [How89b, Fact 1]. For the dual pair (K, M) and any K-type σ, Proposition 1.2 asserts that Hom e ( i,σ) = 0 for some i 0, 1 . Then m(σ, i)=0 and σ / (K, i).  K F ∈ { } Fe ∈ R F For σ (K, ), [How89b] defines the degree deg(σ) with respect to (eG, G′) as the ∈ R F minimal degree of polynomials in the σ-isotypic subspace σ = ϕ∈Hom e (σ,F) Im(ϕ). e F K P Corollary 1.4. If σ (K, i) for some i 0, 1 , then deg(σ) i (mod 2). ∈ R F ∈ { } ≡ Proof. By Corollary 1.3, σe / (K, −i). So σ i and deg(σ) i (mod 2).  ∈ R F1 F ⊆ F ≡ Theorem 1.5. For a real reductivee dual pair (G, G′), let π be an irreducible (g, K)- module with a non-zero theta lift. If σ , σ (K,π), then σ , σ (K, ) and 1 2 1 2 e deg(σ ) deg(σ ) (mod 2). ∈ R ∈ R F 1 ≡ 2 e e Proof. Let Vπ be an underlying space of π. By definition Hom e ( i,Vπ) = 0 for some g,K F 6 i 0, 1 . Take a non-zero T Hom e ( i,Vπ). As π is an irreducible (g, K)-module, ∈ { } ∈ g,K F the image T ( i)= Vπ. F e For j 0, 1 , let Wσ be an underlying space of σj. As dim Hom e (Vπ,Wσ ) = ∈ { } j K j m(σj,Vπ) > 0, take a non-zero ϕj Hom e (Vπ,Wσ ). As Wσ is an irreducible K- ∈ K j j module, the image ϕj(Vπ) = Wσ . Now the surjective ϕj T : i Wσ gives a j ◦ F → j e non-zero element of Hom e ( i,Wσ ). So m(σj, i) > 0, σj (K, i) (K, ), and K F j F ∈ R F ⊆ R F deg(σj) i (mod 2) for both j 0, 1 by Corollary 1.4.  ≡ ∈ { } e e Remark. Corollary 1.3 (and the degree-parity preservation of Theorem 1.5 as a con- sequence) can also be deduced from classical invariant theory, similar to the proof of [Fan17, Lemma 6] based on [How89b, (3.9)(b)] and [How89a].

2. Parity Preservation From Theorem 1.5, we deduce the preservation of parity of all K-types occurring in an arbitrary irreducible (g, K)-module of a Lie group G in real reductive dual pairs. Theorem 2.1 (Parity preservation). Let G and K be as in the following table, with K embedded in G as a maximal compact subgroup in the usual way. If π is an irreducible (g, K)-module and σ1, σ2 (K,π), then ε(σ1)= ε(σ2), where ε : (K) Z/2Z is the parity of K-types defined explicitly∈ R in the next subsection. R → G K G K G K GLm(R) O(m) Op(C) O(p) Sp2n(C) Sp(n) GLm(C) U(m) O(p,q) O(p) O(q) Sp2n(R) U(n) × ∗ GLm(H) Sp(m) Sp(p,q) Sp(p) Sp(q) O (2n) U(n) U(p,q) U(p) × U(q) × PARITY PRESERVATION OF K-TYPES UNDER THETA CORRESPONDENCE 4

2.1. Parametrization and parity for K-types. For K = U(n), T = U(1)n = diag(U(1),...,U(1)) is a maximal torus, with the standard system of positive roots + ∆ (t, k) = ei ej 1 6 i < j 6 n . Write t0 for the real Lie algebra of T . Write each { −∗ | } weight in √ 1t as the n-tuple of coefficients under the basis e , . . . , en. Then a U(n)- − 0 1 type is parametrized by its highest weight (a1, . . . , an) with integers a1 > a2 > > an. For K = Sp(n), T = Sp(1)n = diag(Sp(1),...,Sp(1)) is a maximal torus,· with · · the + standard system of positive roots ∆ (t, k) = ei ej 1 6 i < j 6 n 2ei 1 6 i 6 { ± | } ∪ { | n . Write each weight in √ 1t∗ as the n-tuple of coefficients under the basis e , . . . , } − 0 1 en. Then a Sp(n)-type is parametrized by its highest weight (a1, . . . , an) with integers a > a > > an > 0. 1 2 · · · Lemma 2.2 ([Wey97]). Embed O(n) in U(n) as O(n) = U(n) GL(n, R). There is a ≃ ∩ bijection (O(n)) (b , b , . . . , br, 1,..., 1, 0,..., 0) (U(n)) : 2r+s 6 n, br > 2 , R ←→ { 1 2 ∈ R } s n−r−s such that an O(n)-type σ corresponds| to{z a U}(n|)-type{z }λ if and only if the highest weight vectors of λ generate an O(n)-module equivalent to σ. Therefore, an O(n)-type σ can be parametrized as follows (with [t] the greatest integer less than or equal to t): 6 n (b1, b2, . . . , br, 1,..., 1, 0,..., 0; +1) if r + s 2 ,  s n − −  [ 2 ] r s  n (b , b , . . . , br, 1|,...,{z 1}, |0,...,{z }0 ; 1) if 6 r + s 6 n r.  1 2 − 2 − n− r−s n −  2 [ 2 ] n+r+s  | {z } | {z }  n Remark. When r + s = 2 , these two cases coincide and give the same σ.

An O(n)-type is parametrized as (a1, a2, . . . , ax, 0,..., 0; ǫ) with integers a1 > a2 > n [ 2 ] > ax > 1, ǫ 1 , and corresponding| U(p)-type{z (a , a} , . . . , ax, 1,..., 1 , 0,..., 0). · · · ∈ {± } 1 2 1−ǫ 2 (n−2x) When n is even and n = 2x, the two choices of ǫ 1 give the same| O{z(n})-type. Define the parity ε : (K) Z/2Z for K = U∈(n {±), Sp}(n) or O(n) as: R → K a K-type σ is parametrized as parity ε(σ) Z/2Z ∈ U(n) (a1, . . . , an) with integers a1 > a2 > > an n · · · i=1 ai (mod 2) Sp(n) (a1, . . . , an) with integers a1 > a2 > > an > 0 · · · n P[ 2 ] 1−ǫ (a , . . . , a n ; ǫ) with integers ai + n O(n) 1 [ 2 ] i=1 2 · a > a > > a n > 0, and ǫ 1 P (mod 2) 1 2 · · · [ 2 ] ∈ {± } Remark. The parity of an O(n)-type is the same as that of its corresponding U(p)-type. r For K = K1 Kr with Ki = U(ni), Sp(ni) or O(ni), as (K)= i=1 σi : σi ×···× r Rr { ∈ (Ki) , define the parity ε : (K) Z/2Z by ε( σi)= ε(σi) NZ/2Z. R } R → i=1 i=1 ∈ N P 2.2. Non-vanishing and splitting conditions. To prove Theorem 2.1, we recall the non-vanishing and splitting conditions for the local theta correspondence over R. PARITY PRESERVATION OF K-TYPES UNDER THETA CORRESPONDENCE 5

Let W be a real symplectic . A reductive dual pair (G, G′) in Sp(W ) is called irreducible if G G′ acts irreducibly on W . Each reductive dual pair (G, G′) in Sp(W ) can be decomposed· into a direct sum of irreducible pairs, namely, there is an k ′ orthogonal direct sum decomposition W = i=1 Wi such that G G acts irreducibly on ′ · Wi, and the restrictions of actions of (G, G )L to Wi define irreducible reductive dual pairs ′ ′ ′ ′ (Gi, Gi) in Sp(Wi). Indeed, G = G1 Gk and G = G1 Gk. All irreducible real reductive dual pairs are classified×···× in the following table (c.f.×···× [How79]). Type I Sp Type II Sp (O(p,q),Sp(2n, R)) Sp(2n(p + q), R) (GLm(R), GLn(R)) Sp(2mn, R) (Op(C),Sp(2n, C)) Sp(4pn, R) (GLm(C), GLn(C)) Sp(4mn, R) ∗ (Sp(p,q), O (2n)) Sp(4n(p + q), R) (GLm(H), GLn(H)) Sp(8mn, R) (U(p,q),U(r, s)) Sp(2(p + q)(r + s), R) An irreducible real reductive dual pair (G, G′) of type I is said to be in the stable range with G the smaller member if the defining module of G′ contains an isotropic subspace of the same dimension as that of the defining module of G. All irreducible real reductive dual pairs (G1, G2) in the stable range are listed in following table.

G1 G2 with G1 smaller with G2 smaller Op(C) Sp2n(C) n > p p > 4n O(p,q) Sp2n(R) n > p + q p,q > 2n Sp(p,q) O∗(2n) n > 2(p + q) p,q > n U(p,q) U(r, s) r, s > p + q p,q > r + s

Write the two elements of Ker(Sp Sp) = µ as e and e, such that e = ( e)2 is → 2 − − the identity element. A (g, K)-module is called genuine if e acts on it as the scalar f − multiplication by 1. Clearly, an irreducible (g, K)-module π with a non-zero theta lift e must be genuine, since− ω( e) acts on by the scalar 1. Conversely, two lemmas hold: − F e − Lemma 2.3 (Non-vanishing of theta liftings in the stable range [PP08]). If (G, G′) is an irreducible real reductive dual pair of type I in the stable range with G the smaller member, then each genuine irreducible (g, K)-module has a non-zero theta lift. ′ Lemma 2.4 ([Mœg89, AB95, LPTZ03]). Fore (G, G ) = (GLm(F ), GLn(F )) with F = R, C or H, if n > m, then each genuine irreducible (g, K)-module has a non-zero theta lift. Suppose that the covering G G splits, namely,e there exists an embedding G ֒ G → → such that the composition G ֒ G G is the identity map on G. We may identify G →e → e as a subgroup of G via this embedding. Then G = G µ in the sense that the two e × 2 subgroups G and µ = Ker(G G) commute, generate G, and G µ = e . Similarly e2 → e ∩ 2 { } we have K = K µ . An irreducible (g, K)-module π gives rise to a genuine irreducible × 2 e e (g, K)-moduleπ ˜, with the same underlying space and actions of (g, K), whileπ ˜( e) acts e − by the scalar 1. Similarly, a K-type σ gives rise to a genuine K˜ -typeσ ˜, with the same e underlying space− and actions of K, whileσ ˜( e) acts by the scalar 1. Consider the − − ,actions of K on the Fock model via the embedding K ֒ K = K µ . Clearly F → × 2 σ (K, ) σ˜ (K, ). e ∈ R F ⇐⇒ ∈ R F e PARITY PRESERVATION OF K-TYPES UNDER THETA CORRESPONDENCE 6

In that case we define the degree deg(σ) for σ (K, ) by deg(σ) = deg(˜σ). ∈ R F Proposition 2.5. Let (G, G′) be an irreducible real reductive dual pair, either in the stable range or of type II, with G the smaller member. Suppose that G G splits over G. If π is an irreducible (g, K)-module, and σ , σ (K,π), then σ →, σ (K, ) 1 2 e1 2 and deg(σ ) deg(σ ) (mod 2). ∈ R ∈ R F 1 ≡ 2

Proof. As we said, π gives rise to a genuine irreducible (g, K)-moduleπ ˜, while σ1 and σ (K,π) give rise toσ ˜ andσ ˜ (K, π). By Lemma 2.3, 2.4, θ(˜π) = 0. By 2 ∈ R 1 2 ∈ R e 6 Theorem 1.5,σ ˜ ,σ ˜ (K, ), and deg(˜σ ) deg(˜σ ) (mod 2). Equivalently, we have 1 2 1 e 2 σ , σ (K, ) and∈ deg(R σ F) deg(σ ) (mod≡e 2).  1 2 ∈ R F e 1 ≡ 2

For an irreducible real reductive dual pair (G1, G2), The following table gives some sufficient conditions for Gi Gi to split (c.f. [AB95, AB98, Pau98, Ada07]). → e G G G G splits G G splits 1 2 1 → 1 2 → 2 C C Op( ) Sp2n( ) e alwayse O(p,q) Sp2n(R) if n is even if p + q is even Sp(p,q) O∗(2n) always U(p,q) U(r, s) if r + s is even if p + q is even GLm(R) GLn(R) if n is even if m is even GLm(C) GLn(C) always GLm(H) GLn(H) always

Let (G, G′) be an irreducible real reductive dual pair with G G splitting. The following table lists deg(σ) explicitly for σ (K, ) (c.f. [Mœg89,→ AB95, Pau98, e LPTZ03, Pau05]). ∈ R F

G G′ K σ (K, ) deg(σ) ∈ R F p [ 2 ] − C C p 1 ǫ Op( ) Sp2n( ) O(p) (a1, . . . , a[ 2 ]; ǫ) i=1 ai + 2 (p 2 i : ai > 0 ) C C n − |{ }| Sp2n( ) Op( ) Sp(n) (a1, . . . , an) Pi ai =1p [ 2 ] 1−ǫ O(p) (a , . . . , a p ; ǫ) P a + (p 2 i : a > 0 ) R 1 [ 2 ] i=1 i 2 i O(p,q) Sp2n( ) q − |{ }| [ 2 ] 1−η O(q) (b , . . . , b q ; η) P+ b + (q 2 j : b > 0 ) 1 [ 2 ] j=1 j 2 j × ⊗ n p−q − |{ }| Sp n(R) O(p,q) U(n) (a , . . . , an) P ai 2 1 i=1 | − 2 | ∗ Sp(p) (a1, . . . , ap) Pp q Sp(p,q) O (2n) i=1 ai + j=1 bj Sp(q) (b1, . . . , bq) ∗ × ⊗ Pn P O (2n) Sp(p,q) U(n) (a , . . . , an) ai p + q 1 i=1 | − | U(p) (a1, . . . , ap) Pp r−s q r−s U(p,q) U(r, s) i=1 ai 2 + j=1 bj + 2 U(q) (b1, . . . , bq) | − | | | × ⊗ m P[ 2 ] − P R R m 1 ǫ GLm( ) GLn( ) O(m) (a1, . . . , a[ 2 ]; ǫ) i=1 ai + 2 (m 2 i : ai > 0 ) C C m − |{ }| GLm( ) GLn( ) U(m) (a1, . . . , am) Pi=1 ai H H m | | GLm( ) GLn( ) Sp(m) (a1, . . . , am) Pi=1 ai P PARITY PRESERVATION OF K-TYPES UNDER THETA CORRESPONDENCE 7

2.3. Proof of Theorem 2.1. By Proposition 2.5, it suffices to find a suitable G′ such that (G, G′) is an irreducible real reductive dual pair satisfying three conditions: (1): It is either in the stable range or of type II, with G the smaller member. (2): The covering G G splits. (3): For (G, G′), the→ degree deg(σ) ε(σ) (mod 2) for any σ (K, ). e ≡ ∈ R F We can take G′ according to the following table, which lists explicit sufficient conditions to ensure (1), (2) and (3). G G′ condition (1) condition (2) condition (3)

Op(C) Sp2n(C) n > p Sp2n(C) Op(C) p > 4n O(p,q) Sp n(R) n > p + q 2 n 2 | Sp2n(R) O(p,q) p,q > 2n 2 p + q 4 p q ∗ | | − Sp(p,q) O (2n) n > 2(p + q) O∗(2n) Sp(p,q) p,q > n 2 p q | − U(p,q) U(r, s) r, s > p + q 2 r + s 4 r s | | − GLm(R) GLn(R) n > m 2 n | GLm(C) GLn(C) n > m GLm(H) GLn(H) n > m

2.4. Generalization. Theorem 2.1 can be generalized to all members of (reducible) real reductive dual pairs. Theorem 2.6. Let G be a member of a real reductive dual pair, with a maximal compact subgroup K. If π is an irreducible (g, K)-module, and σ, σ′ (K,π), then ε(σ)= ε(σ′). ∈ R Proof. Any real reductive dual pair is a direct sum of irreducible ones, so G = G 1 × G2 Gr with each Gi a member of an irreducible real reductive dual pair. Then ×···× r K = K1 Kr with Ki a maximal compact subgroup of Gi. By [GK13], π = i=1 πi, ×···× r ′ r ′ where πi is an irreducible (gi, Ki)-module. Moreover, σ = i=1 σi and σ = Ni=1 σi, ′ with σi and σi (Ki,πi). Theorem 2.1 holds for (Gi, Ki)N (up to isomorphisms).N So ′ ∈ R r r ′ ′ ε(σi)= ε(σi) for all i. Therefore, ε(σ)= i=1 ε(σi)= i=1 ε(σi)= ε(σ ).  In the end, please note that the phenomenonP of parityP preservation of K-types in an irreducible admissible representation is well-known to experts for many kinds of Lie groups for many years. Many cases can be proved in a more elementary way, without the help of Howe’s theory of theta correspondence. If G is connected with rank(G) = rank(K) and of type A, C or D (for example G = U(p,q) or Sp(2n, R)), this follows easily from the fact that the difference between any two (highest) weights in an irreducible (g, K)-module is a sum of roots, which are “even” in our sense of parity. However, in other cases, especially when G is disconnected (so that the definition of “parity” is less natural), our approach makes the phenomenon much clearer. This note extends parts of the author’s doctoral thesis work in HKUST. It was written partially while the author was visiting the Institute for Mathematical Sciences, National University of Singapore in 2016. The visit was supported by the Institute. It was completed in Sun Yat-sen University. The author appreciates the referees for suggesting many improvements of the manuscript. PARITY PRESERVATION OF K-TYPES UNDER THETA CORRESPONDENCE 8

References [AB95] Jeffrey Adams and Dan Barbasch, Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 (1995), no. 1, 1–42. MR 1346217 [AB98] , Genuine representations of the , Compositio Math. 113 (1998), no. 1, 23–66. MR 1638210 [Ada07] Jeffrey Adams, The theta correspondence over R, Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 1–39. MR 2401808 [Fan17] Xiang Fan, Explicit induction principle and symplectic-orthogonal theta lifts, J. Funct. Anal. 273 (2017), no. 11, 3504–3548. MR 3706610 [GK13] Dmitry Gourevitch and Alexander Kemarsky, Irreducible representations of a product of real reductive groups, J. Lie Theory 23 (2013), no. 4, 1005–1010. MR 3185208 [How79] Roger Howe, θ-series and invariant theory, Automorphic forms, representations and L- functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 275–285. MR 546602 [How89a] , Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027 [How89b] , Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), no. 3, 535–552. MR 985172 [LPTZ03] Jian-Shu Li, Annegret Paul, Eng-Chye Tan, and Chen-Bo Zhu, The explicit duality correspon- ∗ dence of (Sp(p, q), O (2n)), J. Funct. Anal. 200 (2003), no. 1, 71–100. MR 1974089 [LV80] G´erard Lion and Mich`ele Vergne, The Weil representation, Maslov index and Theta series, Progress in Mathematics, vol. 6, Birkh¨auser, Boston, Mass., 1980. MR 573448 [Mœg89] Colette Mœglin, Correspondance de Howe pour les paires reductives duales: quelques calculs dans le cas archim´edien, J. Funct. Anal. 85 (1989), no. 1, 1–85. MR 1005856 [Pau98] Annegret Paul, Howe correspondence for real unitary groups, J. Funct. Anal. 159 (1998), no. 2, 384–431. MR 1658091 [Pau05] , On the Howe correspondence for symplectic-orthogonal dual pairs, J. Funct. Anal. 228 (2005), no. 2, 270–310. MR 2175409 [PP08] Victor Protsak and Tomasz Przebinda, On the occurrence of admissible representations in the real Howe correspondence in stable range, Manuscripta Math. 126 (2008), no. 2, 135–141. MR 2403182 [Sha62] David Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149–167. MR 0137504 [Wei64] Andr´eWeil, Sur certains groupes d’op´erateurs unitaires, Acta Math. 111 (1964), 143–211. MR 0165033 [Wey97] Hermann Weyl, : their invariants and representations, Princeton Land- marks in Mathematics, Princeton University Press, Princeton, NJ, 1997, Fifteenth printing, Princeton Paperbacks. MR 1488158 E-mail address: [email protected]

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China