## Classification for ${G}_{2}$

Last update: 1 April 2013

## Classification for ${G}_{2}$

The root system $R$ for ${G}_{2}$ has simple roots ${\alpha }_{1}$ and ${\alpha }_{2},$ fundamental weights ${\omega }_{1}$ and ${\omega }_{2},$ and

$⟨α1,α2∨⟩ = -3 ⟨α2,α1∨⟩ = -1, ω1 = 2α1+3α2 ω2 = α1+2α2, and α1 = -ω1+2ω2 α2 = 2ω1-3ω2.$

Irreducible representations.

$Central P(t) Dimension Langlands Indexing character Z(t) parameters triple ta {α1,α2} 1 (ta,∅) (ta,0,1) ∅ 5 (s1ta,{1}) (ta,eα1,1) 5 (s2ta,{2}) (ta,eα2,1) 1 tempered (ta,eα1+eα2,1) tb {α1} 6 (tb,∅) (tb,0,1) ∅ 6 (s1tb,{1}) (tb,eα1,1) tc {α1,α1+3α2} 2 (tc,{2}) (tc,0,1) ∅ 4 (s1tc,{1}) (tc,eα1,1) 4 (s1s2tc,{1}) (tc,eα1+3α2,1) 2 tempered (tc,eα1+eα1+3α2,1) td {α1,α1+2α2} 3 (td,{2}) (td,0,1) ∅ 3 (s1td,{1}) (td,eα1,1) 3 (s2s1s2td,{2}) (td,eα1+2α2,1) 3 tempered (td,eα1+eα1+2α2,1) te {α1,α1+2α2, 3 (te,{2}) (te,0,1) α1+α2,α1+3α2} 1 (s1te,{1}) (te,eα1,1) {α2} 2 (s2s1te,{2}) (te,eα1+α2,1) 1 tempered (te,eα1+eα1+2α2,(21)) 3 tempered (te,eα1+eα1+2α2,(3)) tf {α1,2α1+3α2} 6 (tf,{1}) (tf,0,1) {α1+3α2} 6 (s1tf,{1}) (tf,eα1,1) tg {α1} 6 (tg,{2}) (tg,0,1) {α1+2α2} 6 tempered (tg,eα1,1) th {α2} 6 (th,∅) (th,0,1) ∅ 6 (s2th,{2}) (th,eα2,1) ti {α2,α1+α2} 6 (ti,{1}) (ti,0,1) {α1} 6 (s2ti,{2}) (ti,eα2,1) tj {α2} 6 (tj,{1}) (tj,0,1) {2α1+3α2} 6 tempered (tj,eα2,1) Table 6.1. Irreducible (non principal series) representations$

Table 6.1 lists the irreducible $\stackrel{\sim }{H}\text{-modules}$ by their central characters. We have listed only those central characters $t$ for which the principal series module $M\left(t\right)$ is not irreducible (see Theorem 1.16). The sets $P\left(t\right)$ and $Z\left(t\right)$ are as given in (1.7) and correspond to the choice of representative for the central character displayed in Figure 6.1. The Langlands parameters usually consist of a pair $\left(𝒯,I\right)$ where $I$ is a subset of $\left\{1,2\right\}$ and $𝒯$ is a tempered representation for the parabolic subalgebra ${\stackrel{\sim }{H}}_{I}\text{.}$ In our cases the tempered representation $𝒯$ of ${\stackrel{\sim }{H}}_{I}$ is completely determined by a character $t\in T\text{.}$ Specifically, $𝒯$ is the only tempered representation of ${\stackrel{\sim }{H}}_{I}$ which has $t$ as a weight. In the labeling in Table 6.1 we have replaced the representation $𝒯$ by the weight $t\text{.}$ The notation for the nilpotent elements in the indexing triples is as in Table 6.3.

Table 6.2 lists the irreducible calibrated $\stackrel{\sim }{H}\text{-modules.}$ For each module with central character $t$ we have listed the subset $J\subseteq P\left(t\right)$ such that $\left(t,J\right)$ is the corresponding placed skew shape (see Theorem 1.11). We have listed only those central characters $t$ for which the principal series module $M\left(t\right)$ is not irreducible (see Theorem 1.16).

$Central P(t) Z(t) Dimension Indexing Calibration character triple set J ta {α1,α2} ∅ 1 (ta,0,1) ∅ 5 (ta,eα1,1) {α1} 5 (ta,eα2,1) {α2} 1 (ta,eα1+eα2,1) {α1,α2} tb {α1} ∅ 6 (tb,0,1) ∅ 6 (tb,eα1,1) {α1} tc {α1,α1+3α2} ∅ 2 (tc,0,1) ∅ 4 (tc,eα1,1) {α1} 4 (tc,eα1+3α2,1) {α1+3α2} 2 (tc,eα1+eα1+3α2,1) {α1,α1+3α2} td {α1,α1+2α2} ∅ 3 (td,0,1) ∅ 3 (td,eα1,1) {α1} 3 (td,eα1+2α2,1) {α1+2α2} 3 (td,eα1+eα1+2α2,1) {α1,α1+2α2} te {α1,α1+2α2, {α2} 1 (te,eα1,1) {α1} α1+α2,α1+3α2} 2 (te,eα1+α2,1) {α1,α1+α2} 1 (te,eα1+eα1+2α2,(21)) P(te)\{α1+3α2} th {α2} ∅ 6 (th,0,1) ∅ 6 (th,eα2,1) {α2} Table 6.2. Calibrated irreducible (non principal series) representations$

Figure 6.1 displays the real parts of the central characters in Table 6.1. If $t\in T$ then the polar decomposition $t={t}_{r}{t}_{c}$ determines an element $\nu \in {ℝ}^{n}$ such that ${t}_{r}\left({X}^{\lambda }\right)={e}^{⟨\nu ,\lambda ⟩}$ (see (1.3)). For each central character ${t}_{p}$ the point labeled by $p$ in Figure 6.1 is the graph of the corresponding ${\nu }_{p}\in {ℝ}^{n}\text{.}$ Assume (for pictorial convenience) that $q$ is a positive real number and let

$Hβ= { x∈ℝn ∣ ⟨β,x⟩=0 } ,and Hβ±δ= { x∈ℝn ∣ ⟨β,x⟩= ln(q±2) } ,$

for each positive root $\beta \text{.}$ The dotted lines display the (affine) hyperplanes ${H}_{\beta ±\delta }\text{.}$

$Hα1 Hα2 Hα1+3α2 H2α1+3α2 Hα1+α2 Hα1+2α2 a b c,d,e f g h i j Figure 6.1. Real parts of central characters in Table 6.1$

Tempered and square integrable representations. The irreducible tempered representations with real central character can be indexed by the irreducible representations of the Weyl group $W$of type ${G}_{2}$ (see [BMo1989, p. 34]). Equivalently, these representations can be indexed by the pairs $\left(n,\rho \right)$ which appear in the Springer correspondence. The $n$ and $\rho$ will also be elements of the indexing triple for the corresponding tempered representation of $\stackrel{\sim }{H}\text{.}$ Here $n$ is a nilpotent element of the Lie algebra $𝔤=\text{Lie}\left(G\right),$ $G$ is the complex simple group over $ℂ$ of type ${G}_{2}$ and $\rho$ is an irreducible representation of the component group ${Z}_{G}\left(n\right)/{Z}_{G}{\left(n\right)}^{\circ }$ (see [Car1985]). For each root $\beta \in R$ let ${e}_{\beta }$ be an element of the root space ${𝔤}_{\beta }\text{.}$ The five nilpotent orbits in g and the corresponding tempered representations of $\stackrel{\sim }{H}$ are as in Table 6.3. The notation ${S}_{3}$ denotes the symmetric group on three elements, which has irreducible representations indexed by the partitions (3),(21), $\left({1}^{3}\right)$ of 3. We have used the notation of Carter [Car1985, p.427] to label the irreducible representations of the Weyl group $W$ of type ${G}_{2}\text{.}$

$Nilpotent orbit ZG(n)/ZG(n)∘ Indexing triple Sq. int. W rep. regular 1 (ta,eα1+eα2,1) yes ϕ1,0 subregular S3 (te,eα1+eα1+2α2,(3)) yes ϕ2,1 (te,eα1+eα1+2α2,(21)) yes ϕ1,3′ subminimal 1 (tj,eα2,1) no ϕ2,2 minimal 1 (tg,eα1,1) no ϕ1,3′′ 0 1 (to,0,1) no ϕ1,6 Table 6.3. Tempered representations and the Springer correspondence$

The only other tempered representations are the representations labeled by the triples $\left({t}_{c},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+3{\alpha }_{2}},1\right)$ and $\left({t}_{d},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+2{\alpha }_{2}},1\right)\text{.}$ These representations are square integrable but do not have real central character.

The modules labeled by $\left({t}_{c},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+3{\alpha }_{2}},1\right),$ $\left({t}_{d},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+2{\alpha }_{2}},1\right),$ $\left({t}_{e},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+2{\alpha }_{2}},\left(3\right)\right),$ $\left({t}_{e},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+2{\alpha }_{2}},\left(21\right)\right),$ are the ones constructed by Lusztig in [Lus1983-2] 4.20, 4.19, 4.7 and 4.22 respectively. In Lusztig’s notation these are the stars (see [Lus1983-2, 4.23]) of the modules labeled by the graphs $𝒢\prime \prime ,$ $𝒢\prime ,$ $𝒢$ and $𝒢\prime \prime \prime ,$ respectively.

$ta s2ta s1s2ta s2s1s2ta s1s2s1s2ta s2s1s2s1s2ta s1s2s1s2s1s2ta s1s2s1s2s1ta s2s1s2s1ta s1s2s1ta s2s1ta s1ta tb s2tb s1s2tb s2s1s2tb s1s2s1s2tb s2s1s2s1s2tb s1s2s1s2s1s2tb s1s2s1s2s1tb s2s1s2s1tb s1s2s1tb s2s1tb s1tb tc s2tc s1s2tc s2s1s2tc s1s2s1s2tc s2s1s2s1s2tc s1s2s1s2s1s2tc s1s2s1s2s1tc s2s1s2s1tc s1s2s1tc s2s1tc s1tc td s2td s1s2td s2s1s2td s1s2s1s2td s2s1s2s1s2td s1s2s1s2s1s2td s1s2s1s2s1td s2s1s2s1td s1s2s1td s2s1td s1td te s2te s1s2te s2s1s2te s1s2s1s2te s2s1s2s1s2te s1s2s1s2s1s2te s1s2s1s2s1te s2s1s2s1te s1s2s1te s2s1te s1te tf s2tf s1s2tf s2s1s2tf s1s2s1s2tf s2s1s2s1s2tf s1s2s1s2s1s2tf s1s2s1s2s1tf s2s1s2s1tf s1s2s1tf s2s1tf s1tf tg s2tg s1s2tg s2s1s2tg s1s2s1s2tg s2s1s2s1s2tg s1s2s1s2s1s2tg s1s2s1s2s1tg s2s1s2s1tg s1s2s1tg s2s1tg s1tg th s2th s1s2th s2s1s2th s1s2s1s2th s2s1s2s1s2th s1s2s1s2s1s2th s1s2s1s2s1th s2s1s2s1th s1s2s1th s2s1th s1th ti s2ti s1s2ti s2s1s2ti s1s2s1s2ti s2s1s2s1s2ti s1s2s1s2s1s2ti s1s2s1s2s1ti s2s1s2s1ti s1s2s1ti s2s1ti s1ti tj s2tj s1s2tj s2s1s2tj s1s2s1s2tj s2s1s2s1s2tj s1s2s1s2s1s2tj s1s2s1s2s1tj s2s1s2s1tj s1s2s1tj s2s1tj s1tj Figure 6.2. Calibration graphs for central characters in Table 6.1$

The analysis.

Central characters ${t}_{a},$ ${t}_{b},$ ${t}_{c},$ ${t}_{d},$ ${t}_{h}\text{:}$ The central characters ${t}_{e},$ ${t}_{f},$ ${t}_{g},$ ${t}_{i}$ and ${t}_{j}$ are the only ones which have both $Z\left(t\right)$ and $P\left(t\right)$ nonempty. The other cases are handled by Theorem 1.16 and Theorem 1.11 as in the cases of central characters ${t}_{a}$ ${t}_{b},$ ${t}_{g}$ and ${t}_{o}$ for type ${A}_{2}\text{.}$

The Langlands parameters for each module can be determined from its weight structure. The indexing triple is determined from the Langlands data by using the induction theorem of Kazhdan and Lusztig (see the discussion in [BMo1989, p.34]). Let us give an example to illustrate the procedure. The Langlands parameters $\left({s}_{1}{s}_{2}{t}_{c},\left\{1\right\}\right)$ for the 4 dimensional representation with central character ${t}_{c}$ correspond to the indexing triple $\left({s}_{2}{t}_{c},{e}_{{\alpha }_{1}},1\right)$ which is conjugate to the triple $\left({t}_{c},{s}_{2}{e}_{{\alpha }_{1}},1\right)=\left({t}_{c},{e}_{{\alpha }_{1}+3{\alpha }_{2}},1\right)\text{.}$

The indexing triples for the tempered representations cannot be determined with the use of the Kazhdan-Lusztig induction theorem. The indexing triples for the tempered representations with real central character are determined from the Springer correspondence, see Table 6.3 and [BMo1989, p.34]. The two tempered representations with central characters ${t}_{c}$ and ${t}_{d}$ do not have real central character. By the last two sentences of [Lus1983-2, 2.10] we know that the indexing triples for these representations contain the subregular nilpotent and that the component groups are isomorphic to $ℤ/3ℤ$ and $ℤ/2ℤ$ respectively. In both cases the component group acts trivially on $K\left({ℬ}_{s,u}\right)$ and so $\rho =1\text{.}$ The fact that the elements ${e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+3{\alpha }_{2}}$ and ${e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+2{\alpha }_{2}}$ are representatives of the subregular nilpotent orbit can be derived from the analysis in [Jac1997, Theorem 4.40] or [Stu1971]. This determines the triples $\left({t}_{c},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+3{\alpha }_{2}},1\right)$ and $\left({t}_{d},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+2{\alpha }_{2}},1\right)\text{.}$

Central character ${t}_{e}\text{:}$ Theorem 1.11 applied to the placed skew shapes $\left({t}_{e},\left\{{\alpha }_{1},{\alpha }_{1}+{\alpha }_{2}\right\}\right),$ $\left({t}_{e},\left\{{\alpha }_{1}\right\}\right)$ and $\left({t}_{e},\left\{{\alpha }_{1},{\alpha }_{1}+{\alpha }_{2},{\alpha }_{1}+2{\alpha }_{2}\right\}\right)$ produces, respectively, a two dimensional irreducible module $M$ with $\text{supp}\left(M\right)=\left\{{s}_{2}{s}_{1}{t}_{e},{s}_{1}{s}_{2}{s}_{1}{t}_{e}\right\},$ a one dimensional irreducible module $N$ with $\text{supp}\left(N\right)=\left\{{s}_{1}{t}_{e}\right\}$ and a one dimensional irreducible module ${N}^{*}$ with $\text{supp}\left({N}^{*}\right)=\left\{{s}_{2}{s}_{1}{s}_{2}{s}_{1}{t}_{e}\right\}\text{.}$ Lusztig [Lus1983-2] Theorem 4.7 constructs a 3-dimensional irreducible $\stackrel{\sim }{H}\text{-module}$ $P$ with $\text{dim}\left(\right){P}_{{t}_{e}}^{\text{gen}}=2$ and $\text{dim}\left({P}_{{s}_{1}{t}_{e}}^{\text{gen}}\right)=1\text{.}$ In Lusztig’s notation this is the module labeled by the graph $𝒢$ for ${\stackrel{\sim }{G}}_{2}\text{.}$

As described in [Lus1983-2, 4.23] we can twist the module $P$ by an involutive automorphism of $\stackrel{\sim }{H}$ to obtain another 3-dimensional irreducible module ${P}^{*}$ which has $\text{dim}\left({\left({P}^{*}\right)}_{{s}_{2}{s}_{1}{s}_{2}{s}_{1}{t}_{e}}^{\text{gen}}\right)=2$ and $\text{dim}\left({\left({P}^{*}\right)}_{{s}_{1}{s}_{2}{s}_{1}{s}_{2}{s}_{1}{t}_{e}}^{\text{gen}}\right)=1\text{.}$

All of the modules $M,$ $N,$ $P,$ ${N}^{*},$ ${P}^{*}$ must appear as composition factors of the principal series module $M\left({t}_{e}\right)\text{.}$ By comparing dimensions of weight spaces, any other module $Q$ which appears in a composition series of $M\left({t}_{e}\right)$ must have $\text{supp}\left(Q\right)\subseteq \left\{{s}_{2}{s}_{1}{t}_{e},{s}_{1}{s}_{2}{s}_{1}{t}_{e}\right\}\text{.}$ Theorem 1.11(b) then implies that $Q$ must be isomorphic to $M\text{.}$ Thus Theorem 1.15 implies that $M,$ $N,$ $P,$ ${N}^{*},$ and ${P}^{*}$ are (up to isomorphism) all the irreducible modules with central character ${t}_{e}\text{.}$

The Langlands parameters for each module are determined from its weight structure. The Kazhdan-Lusztig induction theorem allows us to use the Langlands parameters to determine the indexing triples for the modules which are not tempered. Since ${t}_{e}$ is a real central character the indexing triples for the tempered representations can be determined from the Springer correspondence, see Table 6.3. Alternatively, one can get these triples from [Lus1983-2, 2.10] where it is explained that the nilpotent in the indexing triple is subregular, the variety ${ℬ}_{s,u}$ (where $s={t}_{e}$ and $u$ is subregular) consists of three disjoint points and a projective line, and the component group is isomorphic to the symmetric group ${S}_{3}\text{.}$ The symmetric group ${S}_{3}$ acts trivially on the line and permutes the three points, which implies that the line corresponds to $\rho =\left(3\right)$ (trivial representation of ${S}_{3}\text{)}$ and the three points are split between the isotypic components $\rho =\left(3\right)$ and $\rho =\left(21\right)\text{.}$ In this case the standard module ${M}_{s,u,\left({1}^{3}\right)}=0\text{.}$ The projective line in ${ℬ}_{s,u}$ corresponds to the two dimensional weight space (P*)tegen in the module ${P}^{*}\text{.}$

Central character ${t}_{f}\text{:}$ Let $ℂ{v}_{f}$ and $ℂ{v}_{{s}_{1}f}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ given by

$T1vf=qvf, Xα1vf=q2vf, X3α2vf=q-2vf, T1vs1f=-q-1vs1f, Xα1vs1f=q-2vs1f, X3α2vs1f=q4vs1f.$

Let

$M1= IndH∼{1}H∼ (ℂvf) and M2= IndH∼{1}H∼ (ℂvs1f).$

By Lemma 1.17 these modules have weights $\text{supp}\left({M}_{1}\right)=\left\{{s}_{2}{t}_{f},{t}_{f},{s}_{1}{t}_{f},{s}_{2}{s}_{1}{t}_{f}\right\}$ and $\text{supp}\left({M}_{2}\right)=\left\{{s}_{1}{t}_{f},{s}_{2}{s}_{1}{t}_{f},{s}_{1}{s}_{2}{s}_{1}{t}_{f},{s}_{2}{s}_{1}{s}_{2}{s}_{1}{t}_{f}\right\}$ respectively. Both ${M}_{1}$ and ${M}_{2}$ are 6 dimensional.

Let $M$ be any $\stackrel{\sim }{H}\text{-module}$ such that ${M}_{{s}_{2}{t}_{f}}\ne 0\text{.}$ By Lemma 1.19 and Proposition 1.6, $\text{dim}\left({M}_{{t}_{f}}^{\text{gen}}\right)=\text{dim}\left({M}_{{s}_{2}{t}_{f}}^{\text{gen}}\right)\ge 2\text{.}$ Since $⟨{s}_{2}{\alpha }_{1},{\alpha }_{1}^{\vee }⟩=⟨{\alpha }_{1}+3{\alpha }_{2},{\alpha }_{1}^{\vee }⟩\ne 0$ it follows from Lemma 1.19 (b) that $\text{dim}\left({M}_{{s}_{2}{t}_{f}}^{\text{gen}}\right)\ge 1\text{.}$ Then, by Proposition 1.6, $\text{dim}\left({M}_{{s}_{2}{s}_{1}{t}_{f}}^{\text{gen}}\right)\ge 1\text{.}$ Adding these numbers up we see that $\text{dim}\left(M\right)\ge 6\text{.}$ It follows that ${M}_{1}$ is irreducible. An analogous argument can be applied to conclude that ${M}_{2}$ is irreducible.

Central character ${t}_{g}\text{:}$ Let $ℂ{v}_{g}$ and $ℂ{v}_{{s}_{1}g}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ given by

$T1vg=qvg, Xα1vg=q2vg, X2α2vg=q-2vg, T1vs1g=-q-1vs1g, Xα1vs1g=q-2vs1g, X2α2vs1g=q2vs1g.$

Let

$M1= IndH∼{1}H∼ (ℂvg)andM2= IndH∼{1}H∼ (ℂvs1g).$

By Lemma 1.17 these modules have weights $\text{supp}\left({M}_{1}\right)=\left\{{s}_{1}{s}_{2}{t}_{g},{s}_{2}{t}_{g},{t}_{g}\right\}$ and $\text{supp}\left({M}_{2}\right)=\left\{{s}_{1}{t}_{g},{s}_{2}{s}_{1}{t}_{g},{s}_{1}{s}_{2}{s}_{1}{t}_{g}\right\}$ respectively. Both ${M}_{1}$ and ${M}_{2}$ are 6 dimensional.

Let $M$ be any $\stackrel{\sim }{H}\text{-module}$ such that ${M}_{{s}_{1}{s}_{2}{t}_{g}}\ne 0\text{.}$ By Lemma 1.19 and Proposition 1.6, $\text{dim}\left({M}_{{t}_{g}}^{\text{gen}}\right)=\text{dim}\left({M}_{{s}_{2}{t}_{g}}^{\text{gen}}\right)=\text{dim}\left({M}_{{s}_{1}{s}_{2}{t}_{g}}^{\text{gen}}\right)\ge 2\text{.}$ Thus $\text{dim}\left(M\right)\ge 6\text{.}$ It follows that $M$ is irreducible. An analogous argument can be applied to conclude that ${M}_{2}$ is irreducible.

Central character ${t}_{i}\text{:}$ Let $ℂ{v}_{i}$ and $ℂ{v}_{{s}_{2}i}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{2\right\}}$ given by

$T2vi=qvi, Xα1vi=vi, Xα2vi=q2vi, T2vs2i=-q-1vs2i, Xα1vs2i=q6vs2i, Xα2vs2i=q-2vs2i.$

Let

$M1= IndH∼{2}H∼ (ℂvi)andM2= IndH∼{2}H∼ (ℂvs2i).$

An argument similar to that for the central character ${t}_{f}$ shows that ${M}_{1}$ and ${M}_{2}$ are irreducible.

Central character ${t}_{j}\text{:}$ Let $ℂ{v}_{j}$ and $ℂ{v}_{{s}_{2}j}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{2\right\}}$ given by

$T2vj=qvj, X2α1vj=q-6vj, Xα2vj=q2vj, T2vs2j=-q-1vs2j, X2α1vs2j=q6vs2j, Xα2vs2j=q-2vs2j.$

Let

$M1= IndH∼{2}H∼ (ℂvj)andM2= IndH∼{2}H∼ (ℂvs2j).$

An argument similar to that for the central character ${t}_{g}$ shows that ${M}_{1}$ and ${M}_{2}$ are irreducible.

## Notes and References

This is an excerpt of a preprint entitled Representations of rank two affine Hecke Algebras, written by Arun Ram, Department of Mathematics, Princeton University, August 5, 1989.

Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.