Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added some proofs to the last chapter
Added proofs for the theorems which describe the classification of coherent families for sp(2n) and sl(n)
Also fixed a typo: for Mathieus theorems to work we needed to noralize m(λ) by a factor of 2/n
Also fixed a typo in the description of the isomorphism W ≅ Sₙ⋉ (ℤ/2)ⁿ
2 files changed, 129 insertions, 22 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/coherent-families.tex | 142 | 124 | 18 |
| Modified | sections/fin-dim-simple.tex | 9 | 5 | 4 |
diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex @@ -85,7 +85,7 @@ combinatorial counterpart. % TODO: Note we will prove that central characters are also invariants of % coherent families % TODO: Prove this -\begin{proposition}[Mathieu] +\begin{proposition}[Mathieu]\label{thm:coherent-family-has-uniq-central-char} Suppose \(\lambda, \mu \in \mathfrak{h}^*\) are such that \(L(\lambda)\) and \(L(\mu)\) are both bounded and \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). Then \(\chi_\lambda = \chi_\mu\). @@ -93,7 +93,7 @@ combinatorial counterpart. % TODO: Note that if σ_β ∙ λ is not dominant integral then L(σ_β ∙ λ) is % infinite-dimensional and 𝓔𝔁𝓽(L(σ_β ∙ λ)) ≅ 𝓔𝔁𝓽(L(λ)) -\begin{proposition} +\begin{proposition}\label{thm:lemma6.1} Let \(\beta \in \Sigma\) and \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded and \(\lambda(H_\beta) \notin \mathbb{N}\). Then \(L(\sigma_\beta \bullet \lambda) \subset \mExt(L(\lambda))\). @@ -128,8 +128,6 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + % TODO: Note that we need a better set of parameters to the space of weights % such that L(λ) is bounded -% TODO: Prove this. The only part worth proving is the fact this is -% W-equivariant, and this is clear from the isomorphism W ≅ S_n ⋉ (ℤ/2)^n % TODO: Revise the notation for this? I don't really like calling this % bijection m \begin{proposition} @@ -154,6 +152,32 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + m(\lambda)_n\). \end{proposition} +\begin{proof} + The fact \(m : \mathfrak{h}^* \to K^n\) is a bijection is clear from the fact + that \(\{\epsilon_1, \ldots, \epsilon_n\}\) is an orthonormal basis for + \(\mathfrak{h}^*\). Veryfying that \(L(\lambda)\) is bounded if, and only if + \(m(\lambda)_1 > m(\lambda)_2 > \cdots > m(\lambda)_{n - 1} > \pm + m(\lambda)_n\) is also a simple combinatorial affair. + + The only part of the statement worth proving is the fact that \(m\) is an + equivariant map, which is equivalent to showing the map + \begin{align*} + \mathfrak{h}^* & \to K^n \\ + \lambda & + \mapsto (\kappa(\epsilon_1, \lambda), \ldots, \kappa(\epsilon_1, \lambda)) + \end{align*} + is equivariant with respect to the natural action of \(W\) on + \(\mathfrak{h}^*\). But this also clear from the isomorphism \(W \cong S_n + \ltimes (\mathbb{Z}/2\mathbb{Z})^n\), as described in + Example~\ref{ex:sp-weyl-group}: \((\sigma_i, (\bar 0, \ldots, \bar 0)) = + \sigma_{\beta_i}\) permutes \(\epsilon_i\) and \(\epsilon_{i + 1}\) for \(i < + n\) and \((1, (\bar 0, \ldots, \bar 0, \bar 1) = \sigma_{\beta_n}\) flips the + sign of \(\epsilon_n\). Hence \(m(\sigma_{\beta_i} \cdot \epsilon_j) = + \sigma_{\beta_i} \cdot m(\epsilon_j)\) for all \(i\) and \(j\). Since \(W\) + is generated by the \(\sigma_{\beta_i}\), this implies that the required map + is equivariant. +\end{proof} + \begin{definition} We denote by \(\mathscr{B}\) the set of the \(m \in (\sfrac{1}{2} + \mathbb{Z})^n\) such that \(m_1 > m_2 > \cdots > m_{n - 1} > \pm m_n\). We @@ -162,10 +186,6 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + and \(\mathscr{B}^- = \{ m \in \mathscr{B} : m_n < 0\}\). \end{definition} -% TODO: Prove this -% Given the Harish-Chandra theorem and the previous proposition, all its left -% is to show that of m(λ)_n = - m(μ)_n then indeed Ext(L(λ)) = Ext(L(μ)). I -% think this is cavered in Lemma 6.1 of Mathieu \begin{theorem}[Mathieu] Given \(\lambda\) and \(\mu\) satisfying the conditions of Lemma~\ref{thm:sp-bounded-weights}, \(\mExt(L(\lambda)) \cong @@ -175,6 +195,36 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + parameterized by \(\mathscr{B}^+\). \end{theorem} +\begin{proof} + Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\), + so that \(m(\lambda), m(\mu) \in \mathscr{B}\). + + Suppose \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). By + Proposition~\ref{thm:coherent-family-has-uniq-central-char}, \(\chi_\lambda = + \chi_\mu\). It thus follows from the Harish-Chandra Theorem that \(\mu \in W + \bullet \lambda\). Since \(m\) is equivariant, \(m(\mu) \in W \cdot + m(\lambda)\). But the only elements in of \(\mathscr{B}\) in \(W \cdot + m(\lambda)\) are \(m(\lambda)\) and \((m(\lambda)_1, m(\lambda)_2, \ldots, + m(\lambda)_{n-1}, - m(\lambda)_n)\). + + Conversely, if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and \(m(\mu)_n = - + m(\lambda)_n\) then \(m(\mu) = \sigma_{\beta_n} \cdot m(\lambda)\) and \(\mu + = \sigma_n \bullet \lambda\). Since \(m(\lambda) \in \mathscr{B}\), + \(\lambda(H_{\beta_n}) \in \sfrac{1}{2} + \mathbb{Z}\) and thus + \(\lambda(H_{\beta_n}) \notin \mathbb{N}\). Hence by + Proposition~\ref{thm:lemma6.1} \(L(\mu) \subset \mExt(L(\lambda))\) and + \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\). + + For each semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-family + \(\mathcal{M}\) there is some \(m = m(\lambda) \in \mathscr{B}\) such that + \(\mathcal{M} = \mExt(L(\lambda))\). The only other \(m' \in \mathscr{B}\) + which generates the same coherent family as \(m\) is \(m' = \sigma_{\beta_n} + \cdot m\). Since \(m\) and \(m'\) lie in different elements of the partition + \(\mathscr{B} = \mathscr{B}^+ \cup \mathscr{B}^-\), the is a unique \(m'' = + m(\nu) \in \mathscr{B}^+\) -- either \(m\) or \(m'\) -- such that + \(\mathcal{M} \cong \mExt(L(\nu))\). +\end{proof} + \section{Coherent \(\mathfrak{sl}_n(K)\)-families} % TODO: Fix n >= 3 @@ -231,6 +281,7 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + \to \{ \mathfrak{sl}_n\textrm{\normalfont-sequences} \} \\ \lambda & \mapsto + \frac{2}{n} ( \kappa(\epsilon_1, \lambda + \rho), \ldots, @@ -245,6 +296,10 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + \emph{not} ordered, but becomes ordered after removing one term. \end{proposition} +% TODO: Note the normalization constant 2/n is choosen because +% λ(H_β) = 2/n κ(λ, β) and m(λ) is thus uniquely characterized by the fact that +% (λ + ρ)(H_β_i) = m(λ)_i - m(λ)_i+1 + % TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose % union corresponds to condition (i) \begin{definition} @@ -271,7 +326,7 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 + It should then be obvious that\dots -\begin{proposition} +\begin{proposition}\label{thm:arrow-implies-ext-eq} Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that \(m(\lambda) \in \mathscr{B}\) -- and suppose that \(\mu \in \mathfrak{h}^*\) is such that \(m(\mu) \in \mathscr{B}\) and there is an arrow \(m(\lambda) @@ -290,15 +345,8 @@ all \(i \ne j\) or \(m_i = m_j\) for some \(i \ne j\), respectively. Similarly, \(\mathfrak{sl}_n\)-sequence \(m\) integral if \(m_i - m_j \in \mathbb{Z}\) for all \(i\) and \(j\). -% TODO: Add notes on what are the sets W ⋅m ∩ 𝓑 : the connected component of -% a given element is contained in its orbit, but a given orbit may contain -% multiple connected components. When m is regular and integral then its orbit -% is the union of n connected components, but otherwise its orbit is precisely -% its connected component (see Lemma 8.3) -% TODO: Perhaps this could be incorporated into the proof of the following -% theorem? Perhaps it's best to create another lemma for this % TODO: Restate the notation for σ_i beforehand -\begin{lemma} +\begin{lemma}\label{thm:connected-components-B-graph} The connected component of some \(m \in \mathscr{B}\) is given by the following. \begin{enumerate} @@ -377,7 +425,6 @@ all \(i\) and \(j\). % TODO: Perhaps this could be incorporated into the discussion of the lemma % that characterizes the weights of sl(n) whose L is bounded -% TODO: Prove this \begin{theorem}[Mathieu] Given \(\lambda, \mu \notin P^+\) with \(L(\lambda)\) and \(L(\mu)\) bounded, \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and @@ -388,6 +435,65 @@ all \(i\) and \(j\). well as by \(\mathscr{B}^+\). \end{theorem} +\begin{proof} + Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\), + so that \(m(\lambda), m(\mu) \in \mathscr{B}\). + + It is clear from Proposition~\ref{thm:arrow-implies-ext-eq} that if + \(m(\lambda)\) and \(m(\mu)\) lie in the same connected component of + \(\mathscr{B}\) then \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). On the other + hand, if \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) then \(\chi_\lambda = + \chi_\mu\) and thus \(\mu \in W \bullet \lambda\). We now investigate which + elements of \(W \bullet \lambda\) satisfy the conditions of + Lemma~\ref{thm:sl-bounded-weights}. To do so, we describe the set + \(\mathscr{B} \cap W \cdot m(\lambda)\). + + % Great migué + If \(\lambda\) is regular and integral then the only permutations of + \(m(\lambda)\) which lie in \(\mathscr{B}\) are \(\sigma_k \sigma_{k+1} + \cdots \sigma_i \cdot m'\) for \(k \le i\) and \(\sigma_k \sigma_{k-1} \cdots + \sigma_i \cdot m'\) for \(k \ge i\), where \(m'\) is the unique ordered + element of \(W \cdot m(\lambda)\). Hence by + Lemma~\ref{thm:connected-components-B-graph} \(\mathscr{B} \cap W \cdot + m(\lambda)\) is the union of the connected components of the \(\sigma_i \cdot + m'\) for \(i \le n\). On the other hand, if \(\lambda\) is singular or + non-integral then the only permutations of \(m(\lambda)\) which lie in + \(\mathscr{B}\) are the ones from the connected component of \(m(\lambda)\) + in \(\mathscr{B}\), so that \(\mathscr{B} \cap W \cdot m(\lambda)\) is + exactly the connected component of \(m(\lambda)\). + + In both cases, we can see that if \(B(\lambda)\) is the set of the \(m' = + m(\mu) \in \mathscr{B}\) such that \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\) + then \(B(\lambda) \subset \mathscr{B} \cap W \cdot m(\lambda)\) is contain in + a union of connected components of \(\mathscr{B}\) -- including that of + \(m(\lambda)\) itself. We now claim that \(B(\lambda)\) is exactly the + connected component of \(m(\lambda)\). This is already clear when \(\lambda\) + is singular or non-integral, so we may assume that \(\lambda\) is regular and + integral, in which case every other \(\mu \in W \bullet \lambda\) is regular + and integral. + + % Great migué + In this situation, \(m(\mu) \in \mathscr{B}^+\) implies \(\mu(H_{\beta_1}) = + m(\mu)_1 - m(\mu)_2 \in \mathbb{Z}\) is negative. But it follows from + Proposition~\ref{thm:lemma6.1} that for each \(\beta \in \Sigma\) there is at + most one \(\mu \notin P^+\) with \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\) + such that \(\mu(H_\beta)\) is a negative integer -- see Lemma~6.5 of + \cite{mathieu}. Hence there is at most one \(m' \in \mathscr{B}^+ \cap W + \cdot m(\lambda)\). Since every connected component of \(\mathscr{B}\) meets + \(\mathscr{B}^+\) -- see Lemma~\ref{thm:connected-components-B-graph} -- this + implies \(B(\lambda)\) is precisely the connected component of + \(m(\lambda)\). + + Another way of stating this is to say that \(\mExt(L(\lambda)) \cong + \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and \(m(\mu)\) lie in the same + connected component. There is thus a one-to-one correspondance between + \(\pi_0(\mathscr{B})\) and the isomorphism classes of semisimple irreducible + coherent \(\mathfrak{sl}_n(K)\)-families. Since every connected component of + \(\mathscr{B}\) meets \(\mathscr{B}^+\) precisely once -- again, see + Lemma~\ref{thm:connected-components-B-graph} -- we also get that such + isomorphism classes are parameterized by \(\mathscr{B}^+\). +\end{proof} + % TODO: Change this % I don't really think these notes bring us to this conclusion % If anything, these notes really illustrate the incredible vastness of the
diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex @@ -687,7 +687,7 @@ This already allow us to compute some examples of Weyl groups. \end{align*} \end{example} -\begin{example} +\begin{example}\label{ex:sp-weyl-group} Suppose \(\mathfrak{g} = \mathfrak{sp}_{2n}(K)\) and \(\mathfrak{h} \subset \mathfrak{g}\) is as in Example~\ref{ex:cartan-of-sp}. Let \(\epsilon_1, \ldots, \epsilon_n \in \mathfrak{h}^*\) be as in @@ -697,10 +697,11 @@ This already allow us to compute some examples of Weyl groups. \(\epsilon_{i+1}\) for \(i < n\) and \(\sigma_{\beta_n}\) switches the sign of \(\epsilon_n\). This translates to a canonical isomorphism \begin{align*} - W & \isoto S_n \ltimes (\mathbb{Z}/2\mathbb{Z})^n \\ - \sigma_{\beta_i} & \mapsto (\sigma_i, \bar 0) = ((i \; i\!+\!1), \bar 0) \\ - \sigma_{\beta_n} & \mapsto (1, \bar 1) + W & \isoto S_n \ltimes (\mathbb{Z}/2\mathbb{Z})^n \\ + \sigma_{\beta_i} & \mapsto (\sigma_i, (\bar 0, \ldots, \bar 0)) \\ + \sigma_{\beta_n} & \mapsto (1, (\bar 0, \ldots, \bar 0, \bar 1)), \end{align*} + where \(\sigma_i = (i \ i\!+\!1)\) are the canonical transpositions. \end{example} If we conjugate some \(\sigma \in W\) by the isomorphism \(\mathfrak{h}^*