Singular Vectors in Chevalley Basis

Here we give explicit formulae for singular vectors of Verma modules over U_q(G), where G is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots ofGwhich we callstraight roots. In some spe-cial cases we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis ofU_q(G^–), whereG^–is the negative roots subalgebra of G, whose basis was introduced in our earlier work in the caseq= 1. This basis seems more economical than the Poincaré–Birkhoff–Witt–type of basis used by Malikov, Fei-gin, and Fuchs for the construction of singular vectors of Verma modules in the case q= 1. Furthermore this basis turns out to be part of a general basis introduced recently for other reasons by Lusztig forU_q(B^–), whereB^–is a Borel subalgebra ofG.

It is well known [109] that every root may be expressed as the result of the action of an element of the Weyl groupWon some simple root. More explicitly, for any"∈B⁺ we have:

"=w(!u) =s_i

1s_i

2⋅ ⋅ ⋅s_i

u(!v), (2.35)

and consequently

s_" =ws_vw^–1=s_i

1. . .s_i

us_vs_i

u. . .s_i

1, (2.36)

where!vis a simple root; the elementw∈W is written in a reduced form, that is, in terms of the minimal possible number of the (generatingW) simple reflectionss_i≡s_!

i; and the action ofs_!,!∈BonH^∗is given bys_!(+) = +– (+,!^∨)!. The positive root"

is called astraight rootif all numbersi₁,. . .i_u,vin (2.35) are different. Note that there

2.4 Singular Vectors in Chevalley Basis 41

may exist different forms of (2.35) involving other elementsw^󸀠and!_v󸀠; however, this definition does not depend on the choice of these elements. Obviously, any simple root is a straight root. Other easy examples of straight roots are those which are sums of simple roots with coefficients not exceeding 1; that is,"=∑kn_k!k, withn_k= 1 or 0.

All straight roots of the simply laced algebrasA_ℓ,D_ℓ,E_ℓare of this form.

Note that for anyG it is enough to consider roots for whichn_k = 0 for 1̸ ≤ k≤ ℓ. Any other root"^󸀠 may be considered as a root of a complex simple Lie algebraG^󸀠 isomorphic to a subalgebra ofG of rankℓ^󸀠 < ℓ, so that"^󸀠 = ∑kn^󸀠_k!^󸀠kandn^󸀠_k = 0 for̸ 1≤ k≤ ℓ^󸀠(!^󸀠_kbeing the simple roots ofG^󸀠). Thus in the case of the straight roots we shall consider always the case whenu=ℓ– 1, and{i₁,. . .i_u,v}will be a permutation of {1,. . . ℓ}.

In what follows we shall use also the following notion. A root𝛾^󸀠 ∈B⁺is called a subrootof𝛾^󸀠󸀠 ∈B⁺if𝛾^󸀠󸀠–𝛾^󸀠= 0 may be expressed as a linear combination of simple̸ roots with nonnegative coefficients.

In this section we considerU_q(G) when the deformation parameterqis not a non-trivial root of unity. This generic case is very important for two reasons. First, forq= 1 all formulae are valid also for the undeformed case, and most formulae first given in [206] were new at the time also forq= 1 (especially in our basis). Second, the formulae for the case whenqis a root of unity use the formulae for genericqas important input as will be explained in Section 2.7.

We prove a statement which presents results from [206] in one uniform formula.

Proposition 1.LetGbe a complex simple Lie algebra and let!k,1≤k≤ ℓ, be the simple roots of the root systemBofG. Let" = n₁!1+n₂!2+⋅ ⋅ ⋅+n_ℓ!_ℓ, where n_k ∈ ℤ+be a straight root (cf.(2.35)) of the positive root systemB⁺ofG, and m a positive integer. Let +∈ H^∗be such that(2.2)is fulfilled with this choice of"and m, but is not fulfilled for any subroot of". Then the singular vector of the Verma module V⁺corresponding to"

and m is given by: combinations of the basis elements H_iof the Cartan subalgebraH ofG, which can be

computed explicitly in all cases. ◊

TheProofof this statement takes the rest of this section. We first treat the case of the simple roots. Then in the following subsections for all complex simple Lie algebras we give their straight roots with explicit presentations of type (2.35), and then we give explicitly the elementsH̃_i1. . . ̃H_iu.

We start with the case of thesimple roots. Let"=!j; then from the expression (2.3) we have:

v^j,m= (X_j^–)^m⊗v₀. (2.39)

Using (1.19) we obtain:

[X_j⁺, (X^–_j)^m] = (X_j^–)^m–1

m–1

∑

k=0

[H_j– 2k]_q

j =

= (X_j^–)^m–1[m]_q

j[H_j–m+ 1]_q

j.

Ifv^j,mis a singular vector we should haveX⁺_jv^j,m= [X_j⁺, (X^–_j)^m]⊗v₀= (X^–_j)^m–1[m]_q

j[+(H_j) – m+ 1]_q

j ⊗v₀0. Ifq_j = q^(!j,!j)/2is not a root of unity then the last equality gives just condition (2.2). (Note thatX_k⁺v^j,m= 0, fork≠j.)

To check (2.37) we use also formulae involving theq-hypergeometric function₂F^q₁:

2F₁^q(–k,s;s+ 1 –p;q^(p–k)/2) =$p0

[k]![s]!

[k+s]!q^ks/2, k> 0,p≤k,s

2F₁^q(a,b;c;z)≡ ∑

n∈ℤ+

Aq(a+n)Aq(b+n)Aq(c)

Aq(a)Aq(b)Aq(c+n)[n]!zⁿ, (2.40) where for integer arguments theq-Gamma functionAqis defined as:

Aq(m)≐[m– 1]_q! , m∈ ℕ (2.41) 1/Aq(m)≐0 , m∈ ℤ–

Suchq-special functionsare in use from XIX century – for a review see [37].

We turn now to the nonsimple straight roots for the different simple Lie algebras.

2.4.1 U_q(A_ℓ)

LetG =A_ℓ, (!i,!j) = –1 for|i–j|= 1, (!i,!j) = 2$ijotherwise. Then every root" ∈B⁺ is given by" = "in = !i+!i+1+⋅ ⋅ ⋅+!i+n–1, where 1 ≤ i ≤ ℓ, 1 ≤ n≤ ℓ–i+ 1. Note thatevery root is straightsince"i,n=s_i("i+1,n) =s_i⋅ ⋅ ⋅s_i+n–2(!i+n–1) =s_i+n–1⋅ ⋅ ⋅s_i+1(!i) = s_i⋅ ⋅ ⋅s_i+t–1s_i+n–1⋅ ⋅ ⋅ ⋅ ⋅ ⋅s_i+t+1(!i+t) = s_i+n–1⋅ ⋅ ⋅s_i+t+1s_i⋅ ⋅ ⋅s_i+t–1(!i+t), 0 ≤ t ≤ n– 1, where we have demonstrated different forms of (2.35) in this case. ForA_ℓthe highest root is

2.4 Singular Vectors in Chevalley Basis 43

given by!̃ = !1+!2+⋅ ⋅ ⋅+!_ℓ. Thus every root" ∈ B⁺is the highest root of a sub-algebra ofA_ℓ; explicitly"inis the highest root of the subalgebraA_nwith simple roots

!_i,!_i+1,. . .,!_i+n–1. This means that it is enough to give the formula for the singular vec-tor corresponding to the highest root. Thus in formula (2.37) with"=!̃we haven_k= 1, 1≤k≤ ℓ, and for the setsi₁,. . .i_u,vwe obtain from!̃=s₁s₂⋅ ⋅ ⋅s_ts_ℓs_ℓ–1⋅ ⋅ ⋅s_t+2(!t+1) the following:

{i₁,. . .i_ℓ–1;v}={1, 2,. . .,t,ℓ,ℓ– 1,. . .,t+ 2;t+ 1},

̃H_i

s ={

{{

H^s, 1≤s≤t

H^{󸀠ℓ+t+1–s}, t+ 1≤s≤j=ℓ– 1 (2.42)

H^k≡H₁+H₂+⋅ ⋅ ⋅+H_k,H^󸀠k≡H_ℓ+H_ℓ–1+⋅ ⋅ ⋅+H_k. Formula (2.37) for A₂ was given in [198] and for arbitraryA_ℓin [201].

2.4.2 U_q(D_ℓ)

LetG = D_ℓ,ℓ ≥ 4, (!i,!j) = –1 for|i–j|= 1,i,j≠ℓand forij= ℓ(ℓ– 2), (!i,!j) = 2$ij

otherwise. First we note that ifn_ℓ–2+n_ℓ–1+n_ℓ≤2, then the root"is a positive root of a subalgebra ofD_ℓof typeA_n,n<ℓ. Thus it remains to consider straight roots"i∈B⁺ given by"i=!i+!i+1+⋅ ⋅ ⋅+!_ℓ. Note that"iis a root of the subalgebraD_ℓ–i+1with simple roots!i,!i+1,. . .,!_ℓ. This means that in order to account for all roots"iit is enough to consider the root"̃ = "₁ =!₁ +!₂+⋅ ⋅ ⋅+!_ℓ= s₁s₂. . .s_ℓ–3s_ℓ–1s_ℓ(!_ℓ–2) = s_ℓ. . .s₂(!₁)

= s₁s₂. . .s_ℓ–3s_ℓ–1s_ℓ–2(!_ℓ) = s₁s₂. . .s_ℓ–3s_ℓs_ℓ–2(!_ℓ–1). Thus in formula (11) with" = "̃we haven_k= 1, 1≤k≤ ℓ, and for the seti₁,. . .i_u,vwe give only the values corresponding to the first presentation of"̃above, namely, we have:

{i₁,. . .i_ℓ–1;v}={1, 2,. . .,ℓ– 3,ℓ– 1,ℓ;ℓ– 2}, (2.43)

̃H_i

s={

{{

H^s, 1≤s≤ ℓ– 3 H_s+1, s=ℓ– 2,ℓ– 1

2.4.3 U_q(E_ℓ)

LetG = E_ℓ, ℓ = 6, 7, 8, (!i,!i+1) = –1, i = 1,. . .,ℓ– 2 (!3,!_ℓ) = –1, (!i,!j) = 2$ij

otherwise. First we note that ifn₂+n₄+n_ℓ ≤ 2 then the root" is a positive root of a subalgebra of E_ℓ of type A_n, n < ℓ. Analogously, if n₂ + n₄ + n_ℓ = 3 and n₁ +n₅ ≤ 1, the root " is a positive root of a subalgebra ofE_ℓ of typeD_n, n < ℓ. Thus it remains to consider the straight root"̃ = !1+⋅ ⋅ ⋅+!_ℓ= s₁s₂s_ℓs_ℓ–1. . .s₄(!3)

= s_ℓ. . .s₂(a₁) = s₁s₂s_ℓ–1. . .s₄s₃(!_ℓ) = s₁s₂s_ℓs₃. . .s_ℓ–2(!_ℓ–1). Thus in formula (2.37) with " = "̃, we have n_k = 1, 1 ≤ k ≤ ℓ, and for the set i₁,. . .i_u,v we give

only the values corresponding to the first presentation of "̃ above, namely, we have:

{i₁,. . .i_ℓ–1;v}={1, 2,ℓ,ℓ– 1,. . ., 4; 3}, (2.44)

̃H_i

s=

{{ {{ {{ {

H_s, s= 1, 2 H_ℓ, s= 3

H^{󸀠󸀠ℓ+3–s}, s= 4,. . .,ℓ– 1 H^󸀠󸀠k≡H_ℓ–1+. . .+H_k.

2.4.4 U_q(B_ℓ)

G = B_ℓ,ℓ ≥ 2, (!i,!j) = –2 if|i–j| = 1, (!i,!j) = 2$ij(2 –$iℓ) otherwise. The straight roots are of two types:"in = !i+!i+1+⋅ ⋅ ⋅+!i+n–1, 1 ≤ i ≤ ℓ, 1 ≤ n≤ ℓ–i+ 1, and

"^󸀠_i =!i+⋅ ⋅ ⋅+!_ℓ–1+ 2!_ℓ, 1≤i<ℓ. Ifi+n– 1 <ℓthen"inis a positive root of a subalgebra ofB_ℓof typeA_n,n<ℓ(with the scalar products scaled by 2 andqreplaced byq²). Thus we are left with two types of straight roots"i= "i,ℓ+1–i =!i+!i+1+⋅ ⋅ ⋅+!_ℓ, 1≤i<ℓ, and"^󸀠_i. As above it is enough to account for the roots withi = 1. Thus we consider

̃"="1=!1+⋅ ⋅ ⋅+!_ℓ=s₁. . .s_ℓ–1(!_ℓ), and ̃"^󸀠="^󸀠₁=!1+⋅ ⋅ ⋅+!_ℓ–1+ 2!_ℓ=s₁. . .s_ℓ–2s_ℓ(!_ℓ–1) (= s_ℓ. . .s₂(!1)). We note that ("̃,"̃) = 2,"̃^∨ = "̃= 2!^∨₁ +⋅ ⋅ ⋅+ 2!^∨_ℓ–1 +!^∨_ℓ, ("̃^󸀠,"̃^󸀠) = 4,

̃"^󸀠∨= (1/2) ̃"^󸀠=!^∨₁ +⋅ ⋅ ⋅+!^∨_ℓ.

Thus in formula (2.37) with"="̃we haven_k= 1, 1≤k≤ ℓ, and

{i₁,. . .i_ℓ–1;v}={1,. . .,ℓ– 1;ℓ}, H̃_is =H^s,q_is =q², s= 1,. . .,ℓ– 1; (2.45) while for"="̃^󸀠we haven_k= 1 +$kℓ, 1≤k≤ ℓ, and

{i₁,. . .i_ℓ–1;v}={1,. . .,ℓ– 2,ℓ;ℓ– 1}, (2.46)

̃H_is={ {{

H^s, s= 1,. . .,ℓ– 2 H_ℓ, s=ℓ– 1 q_i

s=q^{2–$sℓ–1}. The caseℓ= 2 was given first in [202].

2.4.5 U_q(C_ℓ)

LetG = C_ℓ,ℓ ≥ 3, (C₂ ≅ B₂) , (!i,!j) = –1 if|i–j| = 1 andi,j < ℓ, (!i,!j) = –2 ifij = ℓ(ℓ– 1), (!i,!j) = 2$ij(1 +$iℓ) otherwise. The straight roots are of two types:

2.4 Singular Vectors in Chevalley Basis 45

"in =!i+!i+1+⋅ ⋅ ⋅+!i+n–1, 1≤i≤ ℓ, 1≤ n≤ ℓ–i+ 1, and"^󸀠󸀠i = 2!i+⋅ ⋅ ⋅+ 2!_ℓ–1+!_ℓ, 1≤i<ℓ. Ifi+n– 1 <ℓthen"inis a positive root of a subalgebra ofC_ℓof typeA_n,n<ℓ. Thus we are left with two types of straight roots"_i="_i,ℓ+1–i=!_i+!_i+1+⋅ ⋅ ⋅+!_ℓ, 1≤i<ℓ, and"^󸀠󸀠_i. As above it is enough to account for the roots withi = 1. Thus we consider

̃"="1=!1+⋅ ⋅ ⋅+!_ℓ=s_ℓ. . .s₂(a₁) (=s₁. . .s_ℓ–2s_ℓ(!_ℓ–1)) and ̃"^󸀠󸀠="^󸀠󸀠₁ = 2!1+⋅ ⋅ ⋅+ 2!_ℓ–1+!_ℓ

= s₁. . .s_ℓ–1(!_ℓ). We note that ("̃,"̃) = 2,"̃^∨ = "̃= !^∨₁ +⋅ ⋅ ⋅+!^∨_ℓ–1+ 2!^∨_ℓ, ("̃^󸀠󸀠,"̃^󸀠󸀠) = 4,

̃"^󸀠󸀠∨= (1/2) ̃"^󸀠󸀠=!^∨₁ +⋅ ⋅ ⋅+!^∨_ℓ.

Thus in formula (2.37) with"="̃we haven_k= 1, 1≤k≤ ℓ, and {i₁,. . .i_ℓ–1;v}={ℓ,. . ., 2; 1}, H̃_i

s=H^{󸀠ℓ+1–s}, (2.47) q_is=q^1+$s1, s= 1,. . .,ℓ– 1,

while for"="̃^󸀠󸀠we haven_k= 2 –$kℓ, 1≤k≤ ℓ, and

{i₁,. . .i_ℓ–1;v}={1,. . .,ℓ– 1;ℓ}, H̃_is =H^s, (2.48) q_is=q, s= 1,. . .,ℓ– 1.

2.4.6 U_q(F₄)

LetG = F₄, (!1,!1) = (!2,!2) = 2(!3,!3) = 2(!4,!4) = 4, and (!1,!2) = (!2,!3) = 2(!3,!4) = –2 are the nonzero products between the simple roots. We have straight roots of typeA₂:!1+!2,!3+!4;B₂:!2+!3,!2+ 2!3;B₃:!1+!2+!3,!1+!2+ 2!3;C₃:

!₂+!₃+!₄,!₂+ 2!₃+ 2!₄. Thus we are left with the two roots"̃ =!₁+!₂+!₃+!₄= s₁s₂s₄(!3) and"̃^󸀠 = !1+!2+ 2!3+ 2!4 = s₁s₄s₃(!2). We note that ("̃,"̃) = 2,"̃^∨ = "̃

= 2!^∨₁ + 2!^∨₂+!^∨₃ +!^∨₄, ("̃^󸀠󸀠,"̃^󸀠󸀠) = 4,"̃^󸀠󸀠∨= (1/2)"̃^󸀠󸀠=!^∨₁ +!^∨₂ +!^∨₃+!^∨₄. Thus in formula (2.37) with"="̃, we haven_k= 1, 1≤k≤4, and

{i₁,. . .i₃;v}={1, 2, 4; 3}, (2.49)

̃H_is ={ {{

H^s, s= 1, 2 H₄, s= 3 q_i

s=q^2–$s3, (2.50)

while for"="̃^󸀠we haven_k= 1,k= 1, 2,n_k= 2,k= 3, 4, and {i₁,. . .i₃;v}={1, 4, 3; 2}, q_i

s=q^1+$s1,

̃H_i

s={

{{

H₁, s= 1

H^󸀠6–s, s= 2, 3. (2.51)

2.4.7 U_q(G₂)

LetG = G₂, (!₁,!₁) = 3(!₂,!₂) = –2(!₁,!₂) = 6. The nonsimple straight roots are the two roots"̃ = !1+!2= s₁(!2) and"̃^󸀠󸀠󸀠 = !1+ 3!2 = s₂(!1). We note that ("̃,"̃) = 2,

̃"^∨= ̃"= 3!^∨₁ +!^∨₂, ( ̃"^󸀠󸀠󸀠, ̃"^󸀠󸀠󸀠) = 6, ̃"^{󸀠󸀠󸀠∨}= (1/3) ̃"^󸀠󸀠󸀠=!^∨₁ +!^∨₂. Thus in formula (2.37) with"="̃we haven_k= 1,k= 1, 2, and

{i₁;v}={1; 2}, H̃_i1=H₁, q_i1=q³. (2.52) while for"="̃^󸀠󸀠󸀠we haven₁= 1,n₂= 3, and

{i₁;v}={2; 1}, H̃_i

1=H₂, q_i

1=q. (2.53)

Note that for the nonstraight root"̃^󸀠󸀠 = !1+ 2!2= s₂s₁(!2), ("̃^󸀠󸀠,"̃^󸀠󸀠) = 2,"̃^󸀠󸀠∨ = "̃^󸀠󸀠 = 3!^∨₁ + 2!^∨₂and with condition (2.2) fulfilled form= 1:

[(++1,"̃^󸀠∨) – 1]_q󸀠󸀠

̃" = [3+(H₁) + 2+(H₂) + 4]_q= 0 (2.54)

the formula for the singular vector is given as forB₂andm= 1.

Im Dokument Vladimir K. Dobrev Invariant Differential Operators (Seite 52-58)