Notes for the fourth lecture
by Anna Puskás, for March 10th , 2015.
Paper: Characters of Representations and Paths in h∗R , by Peter Littelmann. (Proceedings of Symposia in Pure Mathematics, Volume 61, 1997, Pages 29-49)
Recall that in the last lectures, we introduced the path model in higher rank, looked at an example,
and a few properties.
Plan for today and later lectures:
• Properties of the root operators
• Properties determine the operators
• A First Character Formula with proof.
1
Remarks on the root operators
The root operators have a few key properties, that are simple to formulate and in fact determine the
operators. We review these.
(i) Moving, stretching, dualizing The operators preserve the length of a path, and move the
endpoint by ±α. They are quasi-inverses of each other. We have eα (∗η) = ∗(fα η) for every
η ∈ Π (and dually, fα (∗η) = ∗(eα η)). Further, they are compatible with stretching: ekα (kη) =
k(eα η) (and similarly for f : fαk (kη) = k(fα η)). (Proving stretching property is simple using the
concatenation property. The higher-rank version of the statement is discussed in the notes for
lecture 4. The statement is trivial for straight paths.)
(ii) α-strings For η ∈ Π let m and n be the maximal number of times eα and fα can be applied
without getting ∅, respectively. Further, let mα be the minimum of the function hα : t 7→ (η(1) |
α∨ ). Then
n − m = (η(1) | α∨ ), m ≤ |mα | < m + 1, n ≤ |(η(1) | α∨ ) − mα | < n + 1.
(iii) Reflections We can define the action of the simple reflection sα on Π. Let η ∈ Π and set
k = (η(1) | α∨ ). Then
sα η := f k η if k > 0 and sα η := e−k η otherwise.
Note that s2α = id, and the restriction of the action to Λ ⊂ Π is the usual action of the simple
reflection on the weight lattice. What is more, it is true that (sα η)(1) = sα (η(1)), where the
action on the left is the one defined here, and the action on the right is the usual one.
1
Remark 1. Let B ⊆ Π be finite, such that B ∪ {∅} is stable under the root operators. Then (iii)
implies that its character
X
CharB :=
eη(1)
η∈B
is W -stable.
(iv) Concatenation Let π = λ1 ∗ · · · ∗ λr be such that λ1 , . . . , λr are integral weights. Set ai :=
(λ1 +· · · λi , α∨ )1 and a0 := 0. Then mα is the minimum of the ai . Fix the minimal p with mα = ap
and the maximal q with mα = aq . If p = 0, then eα π = ∅ and if q = r, then fα π = ∅. Otherwise
for x ≤ min{ai − mα | 0 ≤ i ≤ p − 1} and y ≤ min{ai − mα | q + 1 ≤ i ≤ r} we have:
exα π = λ1 ∗ · · · ∗ (exα λp ) ∗ · · · ∗ λr ,
fαy π = λ1 ∗ · · · ∗ (fαy λq+1 ) ∗ · · · ∗ λr .
Let Π̃ be the set of all piecewise linear paths η, as before, but with a fixed parametrization.
define a ditance on Π̃ using the supremum norm
2
We
d(η, π) := max {η(t) − π(t)}.
t∈[0,1]
It is easy to see that if too paths are close with respect to d(., .), then the functions hα , l and r are
close as well. More precisely:
(v) Continuity The path operators are continuous with respect to d(., .), i.e. ∃c ∈ R (depending
only on g) such that if d(η, π) < y and eα (η), eα (π) 6= ∅ then d(eα (η), eα (π)) < cy, and if
fα (η), fα (π) 6= ∅ then d(fα (η), fα (π)) < cy.
Proposition 2. If {fα0 , e0α | α a simple root} is a set of maps Π → Π∪{∅} satisfying the properties (i)
to (v), then fα = fα0 and eα = e0α for all simple roots. In other words, the properties above determine
the root operators.
Proof. By a rational path we mean a path η ∈ Π such that there is an n ∈ N such that the stretching
nη = λ1 ∗ · · · ∗ λr for some λ1 , . . . , λr ∈ Λ.
The properties (ii) and (v) together imply that the action on Π is a continuous extension of the
action on the rational paths. (For any π ∈ Π, we may find a rational path η ∈ Π such that d(η, π) is
very small, and the integers in (ii) are the same for η and π.)
By (i) and the above, it suffices to check the statement fα = fα0 for rational paths. Let λ ∈ Λ such
that a := (λ, α∨ ) > 0. Then (iii) implies fα0 (λ)a = sα (λ), and if k ≤ a, then (iv) implies
fα0ka (aλ) = (kfα0a (λ)) ∗ ((a − k)λ).
1
2
The paper says ai := (λ1 + · · · λi−1 , α∨ ), but I think we want i in the index.
I.e. we do not consider a path and its reparametrization to be the same.
2
This is because aλ is the concatenation (kλ) ∗ ((a − k)λ). In the notation of (iv), we have a0 = 0,
a1 = ak, a2 = a2 . So mα = 0, q = 0 and min{ai − mα | q + 1 ≤ i ≤ r} = ak. So we may apply (iv)
with ka = y ≤ ka. That gives
fα0ka (aλ) = (fα0ka (kλ)) ∗ (((a − k)λ)).
Here
fα0ka (kλ) = sα (kλ) = kfα0a (λ).
From the stretching property in (i) (with stretching factor a), we get that
k
a−k
0k
fα (λ) =
sα (λ) ∗
λ ,
a
a
and hence fα0k (λ) = fαk (λ).
Let η = λ1 ∗ · · · ∗ λr with some λ1 , . . . , λr ∈ Λ. Then the turning points of the path η are integral
weights, and the local minima (local extrema) of the function hα are integers. t follows from (iv) and
the way fα was defined that this remains true for fα η. Since fα0 k λ = fαk λ it follows by (iv) that the
minima of t 7→ (fα0 η(t), α∨ ) are integers. Using (stretching and) (iv) it follows that
k
k
k
fα0 (η) = (fα0 1 λ1 ) ∗ · · · ∗ (fα0 r λr ) = (fαk1 λ1 ) ∗ · · · ∗ (fαkr λr ) = fαk (η).
For rational paths the statement now follows from stretching.
2
A first character formula
Denote by Π+ ⊂ Π the set of paths η such that Imη is contained in the dominant Weyl chamber C,
+
and let Π+
0 be the set of paths such that Imη is in the interior of C (for t > 0). Let Λ be the set
+
of dominant weights, and recall that (in the fnite dimensional case), ρ ∈ Λ denotes half the sum of
positive roots. If B ⊂ Π is a finite subset such that B ∪ ∅ is stable under the root operators eα , fα , we
have already seen that its character
X
CharB :=
eη(1)
η∈B
is stable under the action of W. We give a formula - a path version of the Weyl character formula that computes this character.
Proposition 3.
(
X
w∈W
sgn(w)ewρ )CharB =
X
(
X
w∈W
η∈B
ρ∗η∈Π+
0
3
sgn(w)ew(ρ+η(1)) ).
Corollary 4. For µ ∈ Λ+ let Vµ be the corresponding irreducible representation of g. Then
X
CharVη(1) .
CharB =
η∈B
ρ∗η∈Π+
0
Proof. We prove the proposition. (Then the corollary follows from the Weyl character formula.)
Since both sides are stable alternating under the Weyl group; so it is sufficient to compare the
coefficients of terms corresponding to dominant weights. Let
Ω := {(w, π) | w ∈ W, π ∈ B, w(1) + π(1) ∈ Λ+ },
we have to prove
X
sgn(w)ew(ρ)+π(1) =
w,π∈Ω
X
eρ+η(1) .
η∈B
ρ∗η∈Π+
0
Let Ω0 be the set of pairs (w, π) ∈ Ω0 such that w is the identity and ρ ∗ π ∈ Π+
0 . There is one of
these for each term on the right hand side. The contribution of Ω0 to the left hand side is equal to
the right hand side. Set Ω0 = Ω \ Ω0 . To prove the proposition we have to show
X
sgn(w)ew(ρ)+π(1) = 0.
(1)
w,π∈Ω0
We will define the involution ϕ : Ω0 → Ω0 such that ϕ(w, π) = ϕ(w0 , π 0 ) has the properties sgn(w) =
−sgn(w0 ) and w(ρ) + π(1) = w0 (ρ) + π 0 (1). The existence of such involution obviously implies (1) and
hence the proposition.
The construction of the involution
Consider (w, π) ∈ Ω0 .
If w is not the identity, then w(ρ) + π(1) ∈ Λ+ implies that the path w(ρ) ∗ π meets a proper face
F of the dominant Weyl chamber at least once. Proper face here means the subset of C stabilized by a
fixed nonempty set of elements of W. To clarify: these have a partial ordering given by inclusion. Also,
we mean meeting at t > 0. If w is the identity, then (w, π) ∈
/ Ω0 implies that w(ρ) ∗ π = ρ ∗ π ∈
/ Π+
0 ; i.
e. w(ρ) ∗ π = ρ ∗ π again meets a proper face F of the dominant Weyl chamber C.
For a proper face F of C denote by Ω0 (F ) the set of pairs (w, π) ∈ Ω0 for which w(ρ) ∗ π meets F as
the last face. That is, w(ρ)∗π meets F, and if t0 ∈ [0, 1] is maximal with the property (w(ρ)∗π)(t0 ) ∈ F,
then (w(ρ) ∗ π)(t0 ) is in the interior of F, and (w(ρ) ∗ π)(t) is in the interior of C for every t0 < t ≤ 1.
The set Ω0 is a disjoint union of the sets Ω0 (F ); hence it is sufficient to define the involution for
such a set Ω0 (F ). Let α be a simple root orthogonal to F. For (w, π) ∈ Ω0 (F ) set n := (w(ρ, α∨ )). Note
that n 6= 0.
4
If n > 0, then the minimum of t 7→ (π(t), α∨ ) is at least −n, since w(ρ) ∗ π meets F for some value
t > 0. It follows that enα (π) 6= ∅ and
sα w(ρ) + enα π(1) = w(ρ) − nα + π(1) + nα = w(ρ) + π(1).
Thus the pair (sα w, enα π) satisfies the properties we want from ϕ(w, π). Further, let t0 ∈ [0, 1] be
maximal with the property that (w(ρ) ∗ π)(t0 ) ∈ F. Observe that then (w(ρ) ∗ π)(t) = (sα w(ρ) ∗ enα π)(t)
for all t ≥ t0 . This is because t0 is the last place the function hα (corresponding to the path w(ρ) ∗ π)
attains the value 0; so the last instance where the minimum is attained is not any later than t0 . Hence
ϕ(w, π) := (sα w(ρ), enα π) ∈ Ω0 (F ).
|n|
Similarly, if n < 0, then (w(ρ) ∗ π)(1) = w(ρ) + π(1) ∈ Λ+ , n ≤ −1 implies that fα (π) 6= ∅.
Further,
sα w(ρ) + fα|n| π(1) = w(ρ) − nα + π(1) + nα = w(ρ) + π(1),
|n|
so again the pair (sα w, fα π) satisfies the properties we want from ϕ(w, π). Since n ≤ −1, 0 ∈
[mα + 1, w(ρ) + π(1)]. (Here mα corresponds to the path w(ρ) ∗ π.) Since t0 is the last point in
which 0 is attained by hα (corresponding to the path w(ρ) ∗ π), the last instance where mα + 1 is
|n|
attained is no later than t0 . Hence sα w(ρ) ∗ fα π agrees with w(ρ) ∗ π for t ≥ t0 . This implies
|n|
ϕ(w, π) := (sα w(ρ), fα π) ∈ Ω0 (F ).
Property (i) (in particular, e and f being quasi-inverses, and sα multiplying n by −1) implies that
ϕ is an involution. This completes the proof.
Remark 5. One may be slightly concerned about the fact that in making the argument about the
paths agreeing for t ≥ t0 , we examined hα and mα corresponding to w(ρ) ∗ π and not π. I claim
that this is all right, because of the following observations. Note that w(ρ) does not meet F after
t = 0. Hence t0 is in the subinterval [1/2, 1], where the value of w(ρ) ∗ π depends on π. Let h̃α , m̃α , t̃
denote the appropriate function, minimum, and interval point corresponding to π. Then hα = n ∗ h̃α ,
mα = n + m̃α , and for t ≥ t0 , (w(ρ) ∗ π)(t) = π(˜(t)) + n.
References
[1] P. Littelmann - A Littlewood-Richardson rule for symmetrizable Kač-Moody algebras Inventiones
Mathematicae 116 (1994), 329-346.
[2] P. Littelmann Paths and Root Operators in Representation Theory Annals of Mathematics
Second Series, 142, No. 3 (Nov., 1995), 499-525.
[3] P. Littelmann Characters of Representations and Paths in h∗R Proceedings of Symposia in Pure
Mathematics, Volume 61, 1997, 29-49
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[4] Y. Sanderson Dimension of Demazure modules for rank 2 affine Lie algebras, thesis Compositio
Mathematica, ...
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