Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of types $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$

Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type $B_{\infty}$, $C_{\infty}$, or $D_{\infty}$. Moreover, via this realization, we obtain an explicit description (in terms of Littlewood-Richardson coefficients) of how tensor products of these crystal bases decompose into connected components when their extremal weights are of nonnegative levels. These results, in types $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$, extend the corresponding results due to Kwon, in types $A_{+\infty}$ and $A_{\infty}$; our results above also include, as a special case, the corresponding results (concerning crystal bases) due to Lecouvey, in types $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$, where the extremal weights are of level zero.

Now, from the argument in §4.2, we also observe that for each λ ∈ P of nonnegative level, there exist λ 0 ∈ E and λ + ∈ P + such that B(λ) ∼ = B(λ 0 )⊗B(λ + ) as U q (g)-crystals. Therefore, by combining the results in the four cases above, we finally obtain our second main result (Theorem 4.19), which yields an explicit description (in terms of Littlewood-Richardson coefficients) of the multiplicity of each connected component B(ν) in B(λ) ⊗ B(µ) for general λ, µ ∈ P of nonnegative levels.
This paper is organized as follows. In §2, we introduce basic notation for infinite rank affine Lie algebras and their quantized universal enveloping algebras. In §3, we first recall standard facts about crystal bases of extremal weight modules, and show the connectedness of (the crystal graph of) the crystal basis B(λ) for λ ∈ P . Then, we give a combinatorial realization of B(λ) as the crystal B(λ) of all LS paths of shape λ. In §4, we describe explicitly how the tensor product B(λ) ⊗ B(µ) ∼ = B(λ) ⊗ B(µ) decomposes into connected components when λ, µ ∈ P are of nonnegative levels, deferring the proof of Proposition 4.12 (used to prove Theorem 4.15) to §5. Finally, in §5, after reviewing tensor product multiplicity formulas in [Ko1], [Ko2], we show Proposition 4.12, thereby completing the proof of Theorem 4.15 (and hence Theorem 4.19).
After having finished writing up this article, we were informed by Jae-Hoon Kwon that in [Kw4], he further obtained a description of how the tensor product B(λ) ⊗ B(µ) decomposes into connected components for λ ∈ P + and µ ∈ −P + , in type A ∞ .
2 Basic notation for infinite rank affine Lie algebras.
2.1 Infinite rank affine Lie algebras. Let g be the infinite rank affine Lie algebra of type B ∞ , C ∞ , or D ∞ (see [Kac,§7.11]), that is, the (symmetrizable) Kac-Moody algebra of infinite rank associated to one of the following Dynkin diagrams: Following [Kac,§7.11], we realize this Lie algebra g as a Lie subalgebra of the Lie algebra gl ∞ (C) of complex matrices (a ij ) i,j∈Z with finitely many nonzero entries as follows. If g is of type B ∞ (resp., C ∞ , D ∞ ), then we define elements x i , y i , h i ∈ gl ∞ (C) for i ∈ I := Z ≥0 by (2.1) (resp., (2.2), (2.3)):        x 0 := E 0,1 + E −1,0 , x i := E i,i+1 + E −i−1,−i for i ∈ Z ≥1 , y 0 := 2(E 1,0 + E 0,−1 ), (2.1) (2.2) (2.3) Here, for each i, j ∈ Z, E i,j ∈ gl ∞ (C) is the matrix with a 1 in the (i, j) position, and 0 elsewhere, and e ′ j := The infinite rank affine Lie algebra g is isomorphic to the Lie subalgebra of gl ∞ (C) generated by x i , y i , h i for i ∈ I; the elements x i , y i for i ∈ I are the Chevalley generators, h := i∈I Ch i is the Cartan subalgebra, and Π ∨ := {h i } i∈I is the set of simple coroots. Let Π := {α i } i∈I ⊂ h * := Hom C (h, C) be the set of simple roots for g. Then we have        α 0 = e 0 , α i = −e i−1 + e i for i ∈ Z ≥1 , if g is of type B ∞ , α 0 = 2e 0 , α i = −e i−1 + e i for i ∈ Z ≥1 , if g is of type C ∞ , α 0 = e 0 + e 1 , α i = −e i−1 + e i for i ∈ Z ≥1 , if g is of type D ∞ . (2.4) Here, for each j ∈ Z ≥0 , we define e j ∈ h * = Hom C (h, C) by: e j , e ′ k = δ jk for k ∈ Z ≥0 , where · , · denotes the natural pairing of h * and h; note that h = j∈Z ≥0 Ce ′ j . Let W = r i | i ∈ I ⊂ GL(h * ) denote the Weyl group of g, where r i denotes the simple reflection corresponding to the simple root α i for i ∈ I. If g is of type B ∞ (resp., C ∞ , D ∞ ), then the following equation (2.5) (resp., (2.6), (2.7)) holds for j ∈ Z ≥0 : Also, for the action of W on h, entirely similar formulas hold in all the cases above, with e replaced by e ′ .
For an integral weight λ ∈ P , we define λ (j) ∈ (1/2)Z for j ∈ Z ≥0 by: λ (j) := λ, e ′ j , or equivalently, λ = λ (0) e 0 + λ (1) e 1 + λ (2) e 2 + · · · ; note that either λ (j) ∈ Z for all j ∈ Z ≥0 , or λ (j) ∈ 1/2 + Z for all j ∈ Z ≥0 . If λ ∈ P , then for n sufficiently large, we have since λ ∈ P is a finite sum of integer multiples of fundamental weights; in this case, we set L λ := λ (n) , and call it the level of the integral weight λ. Note that E ⊂ P is identical to the set of integral weights of level zero, and that if λ ∈ P is a dominant integral weight not equal to 0, then the level L λ of λ is positive.

Quantized universal enveloping algebras and their finite rank subalgebras.
We set P ∨ := i∈I Zh i ⊂ h, and let U q (g) = x i , y i , q h | i ∈ I, h ∈ P ∨ denote the quantized universal enveloping algebra of g over C(q) with integral weight lattice P , and Chevalley generators x i , y i , i ∈ I. Also, let U + q (g) (resp., U − q (g)) denote the positive (resp., negative) part of U q (g), that is, the C(q)-subalgebra of U q (g) generated by x i , i ∈ I (resp., y i , i ∈ I). We have a C(q)-algebra anti-automorphism * : U q (g) → U q (g) defined by: x * i = x i , y * i = y i for i ∈ I. (2.14) Note that U ± q (g) is stable under the C(q)-algebra anti-automorphism * : U q (g) → U q (g).
For m, n ∈ Z ≥0 with n ≥ m, we denote the finite interval m, m + 1, . . . , n in I = Z ≥0 by [m, n]. Also, for n ∈ Z ≥0 , we write simply [n] for the finite interval [0, n] = 0, 1, . . . , n in I = Z ≥0 . Let J be a finite interval in I = Z ≥0 , which is of the form [m, n] for some m, n ∈ Z ≥0 with n ≥ m. We denote by g J the (Lie) subalgebra of g generated by x i , y i , i ∈ J, and h. Then, the set ∆ J := ∆ ∩ i∈J Zα i and ∆ + J := ∆ + ∩ i∈J Zα i are the sets of roots and positive roots for g J , respectively.
Remark 2.2. If g is of type B ∞ (resp., C ∞ , D ∞ ), and n ∈ Z ≥3 , then the Lie subalgebra g [n] of g is a "reductive" Lie algebra of type B n+1 (resp., C n+1 , D n+1 ). Also, if n ∈ Z ≥1 , then the Lie subalgebra g [1, n] of g is a "reductive" Lie algebra of type A n .
Denote by U q (g J ) the C(q)-subalgebra of U q (g) generated by x i , y i , i ∈ J, and q h , h ∈ P ∨ , which can be thought of as the quantized universal enveloping algebra of g J over C(q), and denote by U + q (g J ) (resp., U − q (g J )) the positive (resp., negative) part of U q (g J ), that is, the C(q)-subalgebra of U q (g J ) generated by x i , i ∈ J (resp., y i , i ∈ J). Then it is easily seen that U q (g J ) and U ± q (g J ) are stable under the C(q)-algebra anti-automorphism * : U q (g) → U q (g) given by (2.14).
Let W J denote the (finite) subgroup of W generated by the r i for i ∈ J, which is the Weyl group of g J . An integral weight λ ∈ P is said to be J-dominant (resp., J-antidominant) if λ, h i ≥ 0 (resp., λ, h i ≤ 0) for all i ∈ J. For each integral weight λ ∈ P , we denote by λ J the unique element of W J λ that is J-dominant.
3 Path model for the crystal basis of an extremal weight module.
3.1 Extremal elements. Let B be a U q (g)-crystal (resp., U q (g J )-crystal for a finite interval J in I = Z ≥0 ), equipped with the maps e i , f i : B → B ∪ {0} for i ∈ I (resp., i ∈ J), which we call the Kashiwara operators, and the maps wt : B → P , ε i , ϕ i : B → Z ∪ {−∞} for i ∈ I (resp., i ∈ J). Here, 0 is a formal element not contained in B. For ν ∈ P , we denote by B ν the subset of B consisting of all elements of weight ν.
Definition 3.1 (cf. [Kas3,p. 389]). (1) A U q (g)-crystal B is said to be normal if B, regarded as a U q (g K )-crystal by restriction, is isomorphic to a direct sum of the crystal bases of finite-dimensional irreducible U q (g K )-modules for every finite interval K in I = Z ≥0 .
(2) Let J be a finite interval in I = Z ≥0 . A U q (g J )-crystal B is said to be normal if B, regarded as a U q (g K )-crystal by restriction, is isomorphic to a direct sum of the crystal bases of finite-dimensional irreducible U q (g K )-modules for every finite interval K in I = Z ≥0 contained in J.
If B is a normal U q (g)-crystal, then B, regarded as a U q (g J )-crystal by restriction, is a normal U q (g J )-crystal for every finite interval J in I = Z ≥0 . Also, if B is a normal U q (g)crystal (resp., normal U q (g J )-crystal for a finite interval J in I = Z ≥0 ), then we have for b ∈ B and i ∈ I (resp., i ∈ J); in this case, we set for b ∈ B and i ∈ I (resp., i ∈ J). Also, we can define an action of the Weyl group W (resp., W J ) on B as follows. For each i ∈ I (resp., i ∈ J), define S i : B → B by: Then, these operators S i , i ∈ I, give rise to a unique (well-defined) action S : W → Bij(B) (resp., S : W [n] → Bij(B)), w → S w , of the Weyl group W (resp., W J ) on the set B such that S r i = S i for all i ∈ I (resp., i ∈ J). Here, for a set X, Bij(X) denotes the group of all bijections from the set X to itself.
Definition 3.2. Suppose that B is a normal U q (g)-crystal (resp., U q (g J )-crystal for a finite interval J in I = Z ≥0 ).
(1) An element b ∈ B is said to be extremal (resp., J-extremal) if for every w ∈ W and i ∈ I (resp., w ∈ W J and i ∈ J), (3.2) (2) An element b ∈ B is said to be maximal (resp., J-maximal) if e i b = 0 for all i ∈ I (resp., i ∈ J).
(3) An element b ∈ B is said to be minimal (resp., J-minimal) if f i b = 0 for all i ∈ I (resp., i ∈ J).
Remark 3.3. Suppose that B is a normal U q (g)-crystal.
(1) An element b ∈ B is extremal if and only if there exists m ∈ Z ≥0 such that b is [n]-extremal for all n ∈ Z ≥m .
(2) Let n ∈ Z ≥0 . Since the (unique) [n]-maximal element of the crystal basis of the )-module is [n]-extremal, it follows that an [n]-maximal element of B is [n]-extremal. Consequently, by part (1), a maximal element of B is extremal.
(3) Let J be a finite interval of I = Z ≥0 . By the same reasoning as in part (2), we see that a J-minimal element of B is J-extremal.
3.2 Crystal bases of extremal weight modules. In this subsection, we study some basic properties of crystal bases of extremal weight modules; note that all results in [Kas1]- [Kas4] about extremal weight modules and their crystal bases that we use in this paper remain valid in the case of infinite rank affine Lie algebras.
(1) Let M be an integrable U q (g)-module. A weight vector v ∈ M of weight λ is said to be extremal if there exists a family {v w } w∈W of weight vectors in M satisfying the following conditions: (i) If w is the identity element e of W , then v w = v e = v; (ii) for w ∈ W and i ∈ I such that k := wλ, h i ≥ 0, we have x i v w = 0 and y (k) i v w = v r i w ; (iii) for w ∈ W and i ∈ I such that k := wλ, h i ≤ 0, we have y i v w = 0 and x Here, for i ∈ I and k ∈ Z ≥0 , x (k) i and y (k) i denote the k-th q-divided powers of x i and y i , respectively.
(2) Let J be a finite interval in I = Z ≥0 , and let M be an integrable U q (g J )-module. A weight vector v ∈ M of weight λ is said to be J-extremal if there exists a family {v w } w∈W J of weight vectors in M satisfying the same conditions as (i), (ii), (iii) above, with W replaced by W J , and I by J.
The extremal weight U q (g)-module V (λ) of extremal weight λ is, by definition, the integrable U q (g)-module generated by a single element v λ subject to the defining relation that the v λ is an extremal vector of weight λ (see [Kas3,Proposition 8.2.2] and also [Kas5, §3.1]). We know from [Kas3, Proposition 8.2.2] that V (λ) admits the crystal basis (L(λ), B(λ)), and that B(λ) is a normal U q (g)-crystal. If we denote by u λ ∈ B(λ) the element corresponding to the extremal vector v λ ∈ V (λ) of weight λ, then the element u λ ∈ B(λ) is extremal. (1) If λ ∈ P + , then the extremal weight module V (λ) is isomorphic to the irreducible highest weight U q (g)-module of highest weight λ. Therefore, the crystal basis B(λ) is isomorphic, as a crystal, to the crystal basis of the irreducible highest weight U q (g)-module of highest weight λ.
Let J be a finite interval in I = Z ≥0 . As in the case of V (λ), the extremal weight U q (g J )module V J (λ) of extremal weight λ is, by definition, the integrable U q (g J )-module generated by a single element v λ subject to the defining relation that the v λ is a J-extremal vector of weight λ. We denote by (L J (λ), B J (λ)) the crystal basis of V J (λ).
Remark 3.6. Let J be a finite interval in I = Z ≥0 . Then, results entirely similar to those in Remark 3.5 hold for extremal weight U q (g J )-modules and their crystal bases. If λ is module of highest weight λ (resp., lowest weight λ). Also, for λ ∈ P , we have the crystal basis of the finite-dimensional irreducible U q (g J )-module of highest weight λ J , it follows that the crystal graph of B J (λ) is connected, and #(B J (λ)) ν = 1 for all ν ∈ W J λ.
To prove Proposition 3.7 below, we need to recall the description of the crystal basis B(λ) for λ ∈ P from [Kas3]. Let (L(±∞), B(±∞)) denote the crystal basis of U ∓ q (g). Recall from [Kas1, Proposition 5.2.4] that the crystal lattice L(±∞) of U ∓ q (g) is stable under the C(q)-algebra anti-automorphism * : U q (g) → U q (g) given by (2.14). Furthermore, we know from [Kas2, Theorem 2.1.1] that the C-linear automorphism (also denoted by * ) on L(±∞)/qL(±∞) induced by * : L(±∞) → L(±∞) stabilizes the crystal basis B(±∞), and gives rise to a map * : B(±∞) → B(±∞). Now, we set where T λ := {t λ } is a U q (g)-crystal consisting of a single element t λ of weight λ ∈ P (see [Kas3, Example 1.5.3, part 2]), and then set B := λ∈P B λ . Also, we define a map * : is a U q (g)-subcrystal of B λ , and it is isomorphic, as a U q (g)-crystal, to the crystal basis B(λ); under this isomorphism, the extremal element u λ ∈ B(λ) corresponds to the element where u ±∞ is the element of B(±∞) corresponding to the identity element 1 ∈ U ∓ q (g). Thus, we can identify B(λ) with the U q (g)-subcrystal given by (3.3), and identify We have the following proposition (see also [Kw2,Proposition 3.1] in type A +∞ and [Kw3, Proposition 4.1] in type A ∞ ).
Proposition 3.7. Let λ ∈ P be an integral weight. Then, the crystal graph of the crystal basis B(λ) is connected.
Proof. While the argument in the proof of [Kw2, Proposition 3.1] in type A +∞ (or, of [Kw3,Proposition 4.1] in type A ∞ ) still works in the case of type B ∞ , C ∞ , or D ∞ , we prefer to give a different proof.
For each n ∈ Z ≥0 , let (L [n] (±∞), B [n] (±∞)) denote the crystal basis of U ∓ q (g [n] ) ⊂ U ∓ q (g). We deduce from the definitions that U ∓ q (g [n] ) is stable under the Kashiwara operators e i and f i , i ∈ [n], on U ∓ q (g), and that their restrictions to U ∓ q (g [n] ) are exactly the Kashiwara operators e i and f i , i ∈ [n], on U ∓ q (g [n] ), respectively. Therefore, the crystal lattice ) is identical to the A-submodule of the crystal lattice L(±∞) of U ∓ q (g) generated by those elements of the form: Consequently, the crystal basis B [n] (±∞) of U ∓ q (g [n] ) is identical to the subset of B(±∞) consisting of all elements b of the form: b = Xu ±∞ for some monomial X in the Kashiwara operators e i and f i for i ∈ [n]. Now, we define a subset B λ [n] of B λ by: Then, it is obvious from the tensor product rule for crystals that B λ [n] is a U q (g [n] )-crystal isomorphic to the tensor product )-crystals; namely, as )-crystal. We also remark that for all n ∈ Z ≥0 , and (3.5) Claim. The crystal basis B [n] (λ) of the extremal weight U q (g [n] )-module of extremal weight λ is isomorphic, as a U q (g [n] )-crystal, to B(λ) ∩ B λ [n] .
Proof of Claim. We have a C(q)-algebra anti-automorphism ⋆ : (3.6) As in the case of B(±∞), this C(q)-algebra anti-automorphism ⋆ : induces a map ⋆ : )-crystal, and then define ⋆ : given by (3.7). Also, observe that the C(q)-algebra anti-automorphism ⋆ : U q (g [n] ) → U q (g [n] ) given by (3.6) is identical to the restriction to U q (g [n] ) of the C(q)-algebra anti-automorphism * : U q (g) → U q (g) given by (2.14). Consequently, the map ⋆ : We now complete the proof of Proposition 3.7. Take b 1 , b 2 ∈ B(λ) arbitrarily. We show that there exists a monomial X in the Kashiwara operators e i and f i for i ∈ I such that Xb 1 = b 2 . It follows from (3.5) that there exists n ∈ Z ≥0 such that b 1 and b 2 are both contained in )-crystals by the claim above, we deduce from Remark 3.6 with J = [n] that there exists a monomial X in the Kashiwara operators e i and f i for i ∈ [n] such that Xb 1 = b 2 . This finishes the proof of Proposition 3.7.
Proposition 3.8. Let λ ∈ P be an integral weight. For each w ∈ W , we have B(λ) wλ = S w u λ .
Proof. By using the action of the Weyl group W on B(λ), we are reduced to the case w = e. Now, we show that B(λ) λ = u λ . Let b ∈ B(λ) λ . Then, there exists n ∈ Z ≥0 such that ) λ = 1, which implies that b = u λ , as desired.
Remark 3.9. For a general ξ ∈ P , the set B(λ) ξ is not of finite cardinality.
Conversely, suppose that B(λ) ∼ = B(µ) as U q (g)-crystals. Let b ∈ B(λ) be the element corresponding to u µ ∈ B(µ) under the isomorphism B(λ) ∼ = B(µ) of U q (g)-crystals; note is an extremal element of weight µ, and since wt( is an [n]-maximal element. Now we recall that, by the Claim in the proof of Proposition 3.7 and by Remark 3.6 [n] is isomorphic to the crystal basis of the finitedimensional irreducible U q (g [n] )-module of highest weight λ [n] . Therefore, we deduce that the element S w b is the (unique) [n]-maximal element of B(λ) ∩ B λ [n] of weight λ [n] , and hence that µ [n] = wµ = wt(S w b) = λ [n] ∈ W [n] λ, which implies that λ ∈ W µ. Thus we have proved the proposition.
3.3 Lakshmibai-Seshadri paths and crystal structure on them. In this subsection, following [Li1] and [Li2], we review basic facts about Lakshmibai-Seshadri paths and crystal structure on them; note that all results in [Li1] and [Li2] that we use in this paper remain valid in the case of infinite rank affine Lie algebras. We take and fix an arbitrary (not necessarily dominant) integral weight λ ∈ P .
(2) It is obvious from the definitions that B(wλ) = B(λ) for all w ∈ W .
Let J be a finite interval in I = Z ≥0 . We define an LS path of shape λ for g J as follows. First we define a partial order > J on W J λ, which is entirely similar to the one in Definition 3.11, with W replaced by W J , and ∆ + by ∆ + J = ∆ + ∩ i∈J Zα i . If µ, ν ∈ W J λ are such that µ > J ν, then we denote by dist J (· , ·) the maximal length of all possible chains for (µ, ν) in W J λ. Next, for a rational number 0 < a < 1, we define a-chains in W J λ as in Definition 3.13, with W replaced by W J , dist(· , ·) by dist J (· , ·), and ∆ + by ∆ + J . Finally, we define an LS path of shape λ for g J in the same way as in Definition 3.14, with W replaced by W J . We denote by B J (λ) the set of all LS paths of shape λ for g J .
This proves the lemma.
Combining this fact and Remark 3.6 yields the following isomorphism of U q (g J )-crystals: (3.13) In addition, the element π λ J is contained in B J (λ) = B J (λ J ) by Remark 3.15 (1) and Lemma 3.16, and it is the (unique) maximal element of weight 3.4 Isomorphism theorem. The following theorem gives a combinatorial realization of the crystal basis B(λ) of the extremal weight module V (λ) of extremal weight λ in a unified way that is independent of the level L λ of λ ∈ P ; compare with the related results: [Kw2, Theorem 3.21. Let λ ∈ P be a (not necessarily dominant) integral weight. Then, the crystal basis B(λ) of the extremal weight U q (g)-module V (λ) of extremal weight λ is isomorphic, as a U q (g)-crystal, to the crystal B(λ) consisting of all LS paths of shape λ.
Proof. We use the notation in the proof of Proposition 3.7. It follows from the Claim in the proof of Proposition 3.7 and (3.13) with J = [n] that for each n ∈ Z ≥0 , there exists an )-crystals. We show that the following diagram is commutative for all n ∈ Z ≥0 (see also (3.5) and (3.10)): (3.14) )) is connected. Also, we see from Lemma 3.16 that π λ is contained in B [n] (λ) (and hence in B [n+1] (λ)), and from The commutative diagram (3.14) allows us to define a map Φ : B(λ) → B(λ) as follows: [n] for some n ∈ Z ≥0 (see (3.5)).
By using the definition, we can easily verify that the map Φ : B(λ) → B(λ) is indeed an isomorphism of U q (g)-crystals. Thus we have proved the theorem.
Corollary 3.22. (1) For each λ ∈ P , the crystal B(λ) consisting of all LS paths of shape λ is a normal U q (g)-crystal whose crystal graph is connected.
(2) For each λ ∈ P and w ∈ W , we have Note that the tensor product of normal U q (g)-crystals is also a normal U q (g)-crystal.
Hence, by Corollary 3.22 (1), the tensor product B(λ) ⊗ B(µ) for λ, µ ∈ P is a normal U q (g)-crystal. The following proposition plays a key role in the next section.
Proof. For n ∈ Z ≥0 , let B [n] (π ⊗ η) denote the subset of B(π ⊗ η) consisting of all elements of the form: X(π ⊗ η) for some monomial X in the Kashiwara operators e i and f i for i ∈ [n]; the crystal graph of the U q (g [n] )-crystal B [n] (π ⊗ η) is clearly connected. We claim that there exists an isomorphism Ψ n : Now, it is obvious from the definitions that Furthermore, in exactly the same way as in the proof of Theorem 3.21, we obtain the following commutative diagram: This commutative diagram allows us to define a map Ψ : B(π ⊗ η) → B(ν) as follows: for b ∈ B(π ⊗ η), By using the definition, we can easily verify that the map Ψ is an isomorphism of U q (g)crystals. Thus we have proved the proposition.

Decomposition of tensor products into connected components.
In this section, we consider the decomposition (into connected components) of the tensor product B(λ) ⊗ B(µ) for λ, µ ∈ P with L λ , L µ ≥ 0; in fact, it turns out that each connected component is isomorphic to B(ν) for some ν ∈ P . Our main aim is to give an explicit description of the multiplicity m ν λ, µ of a connected component B(ν) for ν ∈ P in this decomposition. It should be mentioned that our results in this section can be regarded as extensions of the corresponding results in [Kw2] (in type A +∞ ) and [Kw3] (in type A ∞ ) to the cases of type B ∞ , C ∞ , D ∞ ; see also [Le] for the case λ, µ ∈ E (in all types A +∞ , B ∞ , C ∞ , D ∞ ).
Combining Propositions 3.23 and 4.3, we conclude that each connected component of is a complete set of representatives for W -orbits in E. We will prove that for each ν ∈ P (2), we may and do assume that λ = λ † ∈ P  (1) Let ν ∈ P is isomorphic, as a U q (g)-crystal, to a connected component of B(λ) ⊗ B(µ), then we have |ν| = |λ| + |µ|.
Since n − p + 1 > 0 by condition (i), we have w i λ [n] , h i = λ [n] , w −1 i h i ≤ 0 in all the cases above. This proves the claim.
Claim. Let β ∈ ∆ be a root for g.
(1) If β is a positive root, then we have µ, β ∨ ≥ 0 since µ ∈ P + . Therefore, we may assume that β is a negative root; note that the set ∆ − of negative roots is as follows (see (2.8)-(2.10)): (4.14) The assumption ξ, β ∨ > 0 now implies that β is of the form: e j −e i for some q ≤ j ≤ q+p−1 and i ≥ q + p. Since µ (j) = L µ for all j ≥ q by the definition of q, it follows that µ, β ∨ = 0.
This proves part (1) of the claim.
By an argument entirely similar to the one for [AK, Lemma 1.6 (1)], we deduce, using the claim above, that S w (π ξ ⊗ π µ ) = π wξ ⊗ π wµ for all w ∈ W (the proof proceeds by induction on the length of w ∈ W with the help of Corollary 3.22 (2)). From this, it easily follows that π ξ ⊗ π µ is extremal. This completes the proof of Theorem 4.6.
If p = 0, then this part is omitted.
Proof. The "if" part follows immediately from Corollary 3.22 (3). We show the "only if" part. As in the proof of Theorem 4.6, we set q m := min j ∈ Z ≥0 | µ (j) m = L µm and p m := # Supp(λ m ) for m = 1, 2. Also, for m = 1, 2, let ξ m ∈ W λ m be the unique element of W λ m such that Then, the proof of Theorem 4.6 shows that for m = 1, 2; observe that if we set ν m := ξ m + µ m for m = 1, 2, then we have If qm = 0, then this part is omitted.
It remains to show that S w (π ⊗ η) is identical to S x (π ⊗ η). If we set ξ := wt(S w (π ⊗ η)), then ξ ∈ P . Therefore, by the bijectivity of the map Θ n : P  such that uν [n+1] = ν [n+1] and w = xu; note that u is equal to a product of the r i 's for i ∈ [n + 1] such that ν [n+1] , h i = 0. Consequently, by using (3.1), we obtain This completes the proof of the proposition.
Let n ∈ Z ≥0 be such that n > p+(N +1)q. Let B Also, we see by Lemmas 4.9 and 4.10 that Let ν ∈ P max, ν is bijective.
The proof of this proposition is given in §5. In the rest of this subsection, we take and fix an arbitrary m ∈ Z ≥0 such that m > p + (N + 1)q. Proof. We show that π ⊗ η is [n]-extremal for all n ≥ m (see Remark 3.3 (1)). We set ν := wt(π ⊗ η) ∈ P Since ν [n] ∈ P [n] + (λ, µ) as seen above, Proposition 4.12 asserts that the map is bijective. Now, fix n ∈ Z ≥0 such that n ≥ m, and set y n := x m x m+1 · · · x n−2 x n−1 ; note that y n ∈ W [n] . Then, the argument above shows that the composite S yn = S xm S x m+1 · · · S x n−2 S x n−1 yields a bijective map from B max, ν as follows: Consequently, the element S y −1 n (π ⊗ η) is contained in B [n] max, ν [n] , and hence is an [n]-maximal element. Because B(λ) ⊗ B(µ) is a normal U q (g)-crystal, it follows from Remark 3.3 (2) that S y −1 n (π ⊗ η) is [n]-extremal, and hence so is π ⊗ η. This proves the corollary. Proof. Let π ⊗ η ∈ B(λ) ⊗ B(µ), and take n ∈ Z ≥0 , with n ≥ m, such that π ⊗ η ∈ B [n] (λ) ⊗ B [n] (µ) (see (3.10)). From Remark 3.20, we see that each connected component of )-crystal, to the crystal basis of a finite-dimensional irreducible U q (g [n] )-module. Therefore, there exists a monomial X in the Kashiwara operators e i for i ∈ [n] such that X(π ⊗ η) ∈ B [n] max . Let ξ ∈ P be the weight of X(π ⊗ η); note that ξ ∈ P [n] + (λ, µ) by Lemma 4.9. We set it is clear that ξ = ν [n] by Lemma 4.10. Then, the argument in the proof of Corollary 4.13 shows that there exists y ∈ W [n] such that S y −1 yields a bijective map from B [n] onto B [m] max, ν . In particular, we have S y −1 X(π ⊗ η) ∈ B [m] max, ν ⊂ B  Theorem 4.15. Let λ ∈ P + and µ ∈ E. We take m ∈ Z ≥0 such that m > p + (N + 1)q as above. Then, we have the following decomposition into connected components : where for each ν ∈ P Remark 4.16. Recall from Remark 2.2 that if g is of type B ∞ (resp., C ∞ , D ∞ ), then g [m] is a "reductive" Lie algebra of type B m+1 (resp., C m+1 , D m+1 ); note that m > p + (N + 1)q ≥ 2.
Furthermore, we know from Remark 3.20 that B [m] (λ) (resp., B [m] (µ)) is isomorphic, as a U q (g [m] )-crystal, to the crystal basis of the finite-dimensional irreducible U q (g [m] )-module for each ν ∈ P [m] + (λ, µ). In particular, the number of those elements ν ∈ P max is extremal by Corollary 4.13, we see from Proposition 3.23 that B is isomorphic, as a U q (g)-crystal, to B(ν).
We have the following (see [  (1) We have the following decomposition into connected components : (4.23) In particular, each connected component of B(λ) ⊗ B(µ) is isomorphic, as a U q (g)-crystal, to B(ν) for some ν ∈ P + .
Remark 4.18. Recall from Remark 2.2 that if g is of type B ∞ (resp., is a "reductive" Lie algebra of type B n+1 (resp., C n+1 , D n+1 ). Furthermore, we know from Remark 3.20 that B [n] (λ) (resp., B [n] (µ)) is isomorphic, as a U q (g [n] )-crystal, to the crystal basis of the finite-dimensional irreducible U q (g [n] )-module of highest weight λ [n] = λ (resp., . Therefore, it follows from part (2) of Theorem 4.17, together with the result in [Li2,§10], that for each ν = ν [n] ∈ P + such that L ν = L λ + L µ , the multiplicity m ν λ, µ in the decomposition (4.23) is equal to the tensor product multiplicity of the corresponding finitedimensional irreducible U q (g [n] )-modules; see §5.2 for an explicit description of this tensor product multiplicity in terms of Littlewood-Richardson coefficients. In particular, we have m ν λ, µ < ∞ for all ν ∈ P + . However, in the case λ, µ ∈ P + , the total number of connected components of B(λ) ⊗ B(µ) is infinite in general; compare with the other cases, in which the total number of connected components of B(λ) ⊗ B(µ) is finite (see Remarks 4.2 and 4.16, and Theorem 4.6). For example, let λ = µ := e 1 + e 2 + e 3 + · · · .
4.5 The general case. Finally, in this subsection, we consider the decomposition (into connected components) of the tensor product B(λ) ⊗ B(µ) for general λ, µ ∈ P such that L λ , L µ ≥ 0. Define λ + , µ + ∈ P + and λ 0 , µ 0 ∈ E as in Remark 4.7. Then we have as U q (g)-crystals. Since λ + ∈ P + and µ 0 ∈ E, it follows from Theorem 4.15 that as U q (g)-crystals, where we take m ∈ Z ≥0 (sufficiently large) as in §4.3, with λ replaced by λ + , and µ by µ 0 . If we define ξ + ∈ P + and ξ 0 ∈ E for each ξ ∈ P [m] + (λ + , µ 0 ) as in Remark 4.7, then we have as U q (g)-crystals. Since λ 0 , ν 0 ∈ E, it follows from Theorem 4.1 that Also, since ξ + , µ + ∈ P + , it follows from Theorem 4.17 that Combining these, we find that Theorem 4.19. Let λ, µ ∈ P be integral weights of nonnegative levels ; namely, L λ , L µ ≥ 0. Then, we have the following decomposition into connected components : where for each ν ∈ P/W , the multiplicity m ν λ,µ is given as follows : 5 Proof of Proposition 4.12. 5.1 Basic notation for Young diagrams. Recall that P denotes the set of all partitions; we usually identify a partition ρ ∈ P with the corresponding Young diagram, which is also denoted by ρ (see, for example, [F]). For ρ, κ ∈ P, we write ρ ⊂ κ if the Young diagram of ρ is contained in the Young diagram of κ; in this case, we denote by κ/ρ the skew Young diagram obtained from the Young diagram of κ by removing that of ρ, and by |κ/ρ| the total number of boxes in this skew Young diagram, i.e., |κ/ρ| = |κ| − |ρ|. It is well-known that for ρ, κ, ω ∈ P, the Littlewood-Richardson coefficient LR ω ρ, κ is nonzero only if ρ ⊂ ω and κ ⊂ ω. Also, the conjugate of the partition ρ ∈ P is denoted by t ρ. In what follows, for L ∈ Z ≥0 and ρ ∈ P, let ι L (ρ) denote the partition whose Young diagram is obtained from the Young diagram of ρ by inserting one row with exactly L boxes between an appropriate pair of adjacent rows of the Young diagram of ρ in such a way that the resulting diagram is also a Young diagram. By convention, we set ι L (ρ) = ρ if L = 0.
We have the following formulas by [Ko1,(1.2.3) and (1.2.2)]: 5.3 Proof of Proposition 4.12. The following proposition plays an essential role in the proof of Proposition 4.12.
Proposition 5.2. Let L ∈ Z ≥0 , and n ∈ Z ≥3 . Let ρ 1 , ρ 2 , ρ ∈ P be partitions satisfying the following conditions : (i) The lengths of these partitions are all less than or equal to n + 1.
(ii) The first part of ρ 1 is equal to L.
Proof of Claim 1. We give a proof only for part (1), since the proof of part (2) is similar. Let u ∈ Z ≥1 be such that the u-th part of ρ is equal to L, and such that the (u + 1)-st part of ρ is less than L; note that u ≥ y > |ρ 2 | by assumption (iii). It follows that u and u − y are equal to the L-th part and the (L + 1)-st part of the conjugate partition t ρ of ρ, respectively (see the figures below). Let τ be the finite permutation of the set Z ≥1 such that b J, n τ (1) > b J, n τ (2) > · · · > b J, n τ (j) > b J, n τ (j+1) > · · · .
Let ω 1 ∈ R(ρ 1 , ω 2 ). Since the first part of ρ 1 is equal to L by assumption (ii), we see that the first part of ω 1 is less than or equal to L. Therefore, the Young diagrams of κ 1 = ι L (ρ 1 ) (resp., ι L (ω 1 )) is obtained by adding one row having exactly L boxes just above the top row of the Young diagram of ρ 1 (resp., ω 1 ). Hence it is obvious that ι L (ω 1 ) ∈ R(κ 1 , ω 2 ).
This completes the proof of Proposition 5.2.