Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added further results on Weyl groups
Added some notes on the computation of the Weyl group for sl(n) and sp(2n), which are necessary for the chapter on coherent families
3 files changed, 128 insertions, 59 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/coherent-families.tex | 46 | 21 | 25 |
| Modified | sections/fin-dim-simple.tex | 139 | 106 | 33 |
| Modified | sections/introduction.tex | 2 | 1 | 1 |
diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex @@ -16,7 +16,7 @@ In addition, it turns out that very few simple Lie algebras admit cuspidal modules at all. Specifically\dots -% TODO: Add sp(2n) to the list of simple Lie algebras! +% TODOO: Add sp(2n) to the list of simple Lie algebras! \begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal} Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose there exists a cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s} \cong @@ -92,24 +92,22 @@ combinatorial counterpart. Then \(\chi_\lambda = \chi_\mu\). \end{proposition} -% TODO: State Lemma 6.1 of Mathieu +% TODOO: State Lemma 6.1 of Mathieu +% TODOO: Treat the case of sl(2) here? \section{Coherent \(\mathfrak{sp}_{2n}(K)\)-families} -% TODO: Fix n >= 2: sp_2 = sl_2 +% TODO: Fix n >= 2 +% TODO: Does the analysis in here work for n = 1? -% TODOO: Note this beforehand -% TODOO: Note beforehand that the Weyl group of sp(2n) is S_n ⋉ (ℤ/2)^n. Write -% down the isomorphism explicitly in terms of the basis Σ -% TODOO: Perhaps its best to keep this information in here? -We can find an orthonormal basis \(\{\epsilon_1, \ldots, \epsilon_n\}\) for -\(\mathfrak{h}^*\) such that \(\Delta = \{\pm \epsilon_i \pm \epsilon_j\}_{i -\ne j} \cup \{2 \epsilon_i\}_i\). Take the basis \(\Sigma = \{ \beta_1, \cdots, -\beta_n \}\) for \(\Delta\) given by \(\beta_n = 2 \epsilon_n\) and \(\beta_i = -\epsilon_i - \epsilon_{i + 1}\) for \(i < n\). +Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sp}_{2n}(K)\) +of diagonal matrices, as in Example~\ref{ex:cartan-of-sp}, and the basis +\(\Sigma = \{\beta_1, \ldots, \beta_n\}\) for \(\Delta\) given by \(\beta_i = +\epsilon_i - \epsilon_{i+1}\) for \(i < n\) and \(\beta_n = 2 \epsilon_n\). +Here \(\epsilon_i : \mathfrak{h} \to K\) is the linear functional which yields +the \(i\)-th entry of the diagonal of a given matrix, as described in +Example~\ref{ex:sp-canonical-basis}. -% TODO: Prove this? This is the core of the classification for sp(2n), but it -% is profoundly technical \begin{lemma}\label{thm:sp-bounded-weights} Then \(L(\lambda)\) is bounded if, and only if \begin{enumerate} @@ -166,14 +164,13 @@ We can find an orthonormal basis \(\{\epsilon_1, \ldots, \epsilon_n\}\) for % TODO: Fix n >= 3 -% TODOO: Add notes about this basis beforehand -Take the standard Cartan subalgebra \(\mathfrak{h} = \{ X \in -\mathfrak{sl}_n(K) : X \ \text{is diagonal}\}\) of \(\mathfrak{sl}_n(K)\) as in -Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i, -\ldots, \epsilon_n \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the -\(i\)-th entry of the diagonal of \(H\). Let \(\Sigma = \{ \beta_1, \ldots, -\beta_{n-1} \}\) be the basis for \(\Delta\) given by \(\beta_i = \epsilon_i - -\epsilon_{i + 1}\). +Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sl}_n(K)\) of +diagonal matrices, as in Example~\ref{ex:cartan-of-sl}, and the basis \(\Sigma += \{\beta_1, \ldots, \beta_{n-1}\}\) for \(\Delta\) given by \(\beta_i = +\epsilon_i - \epsilon_{i+1}\) for \(i < n\). Here \(\epsilon_i : \mathfrak{h} +\to K\) is the linear functional which yields the \(i\)-th entry of the +diagonal of a given matrix, as described in +Example~\ref{ex:sl-canonical-basis}. % TODO: Add some comments on the proof of this: while the proof that these % conditions are necessary is a purely combinatorial affair, the proof of the @@ -206,7 +203,6 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i, % bijection m % TODO: Note that this prove is similar to the previous one, and that the % equivariance of the map follows from the nature of the isomorphism W ≅ S_n -% TODOO: Describe this isomorphism beforehand \begin{proposition} The map \begin{align*} @@ -272,14 +268,14 @@ It should then be obvious that\dots regular. \end{definition} -% TODO: Add notes on what are the sets W . m ∩ 𝓟 : the connected component of +% TODO: Add notes on what are the sets W ⋅m ∩ 𝓟 : 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 -% TODOO: Define the notation for σ_i beforehand +% TODO: Restate the notation for σ_i beforehand \begin{proposition} The connected component of some \(m \in \mathscr{P}\) is given by the following.
diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex @@ -109,6 +109,11 @@ We have already seen some concrete examples. Namely\dots matrices is a Cartan subalgebra of \(\mathfrak{sl}_n(K)\). \end{example} +\begin{example}\label{ex:cartan-of-sp} + It is easy to see from Example~\ref{ex:sp2n} that \(\mathfrak{h} = \{X \in + \mathfrak{sp}_{2n}(K) : X\ \text{is diagonal} \}\) is a Cartan subalgebra. +\end{example} + \begin{example}\label{ex:cartan-direct-sum} Let \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\) be Lie algebras and \(\mathfrak{h}_i \subset \mathfrak{g}_i\) be Cartan subalgebras. Then @@ -485,6 +490,31 @@ called \emph{basis}. \(\alpha = \pm(k_1 \beta_1 + \cdots + k_r \beta_r)\). \end{definition} +\begin{example}\label{ex:sl-canonical-basis} + Suppose \(\mathfrak{g} = \mathfrak{sl}_n(K)\) and \(\mathfrak{h} \subset + \mathfrak{g}\) is the subalgebra of diagonal matrices, as in + Example~\ref{ex:cartan-of-sl}. Consider the linear functionals \(\epsilon_1, + \ldots, \epsilon_n \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the + \(i\)-th entry of the diagonal of \(H\). As observed in + section~\ref{sec:sl3-reps} for \(n = 3\), the roots of \(\mathfrak{sl}_n(K)\) + are \(\epsilon_i - \epsilon_j\) for \(i \ne j\) -- with root vectors given by + \(E_{i j}\) -- and we may take the basis \(\Sigma = \{\beta_1, \ldots, + \beta_{n-1}\}\) with \(\beta_i = \epsilon_i - \epsilon_{i+1}\). +\end{example} + +\begin{example}\label{ex:sp-canonical-basis} + Suppose \(\mathfrak{g} = \mathfrak{sp}_{2n}(K)\) and \(\mathfrak{h} \subset + \mathfrak{g}\) is the subalgebra of diagonal matrices, as in + Example~\ref{ex:cartan-of-sp}. Consider the linear functionals \(\epsilon_1, + \ldots, \epsilon_n \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the + \(i\)-th entry of the diagonal of \(H\). Then the roots of + \(\mathfrak{sp}_{2n}(K)\) are \(\pm \epsilon_i \pm \epsilon_j\) for \(i \ne + j\) and \(\pm 2 \epsilon_i\) -- see \cite[ch.~16]{fulton-harris}. In this + case, we may take the basis \(\Sigma = \{\beta_1, \ldots, \beta_n\}\) with + \(\beta_i = \epsilon_i - \epsilon_{i+1}\) for \(i < n\) and \(\beta_n = 2 + \epsilon_n\). +\end{example} + The interesting thing about basis for \(\Delta\) is that they allow us to compare weights of a given \(\mathfrak{g}\)-module. At this point the reader should be asking himself: how? Definition~\ref{def:basis-of-root} isn't exactly @@ -512,6 +542,13 @@ induces an order in \(Q\), where elements are ordered by their respectively. \end{definition} +\begin{example} + If \(\mathfrak{g} = \mathfrak{sl}_3(K)\) and \(\Sigma\) is as in + Example~\ref{ex:sl-canonical-basis} then the partition \(\Delta^+ \cup + \Delta^-\) induced by \(\Sigma\) is the same as the one described in + section~\ref{sec:sl3-reps}. +\end{example} + \begin{definition}\index{Lie subalgebra!Borel subalgebra}\index{Lie subalgebra!parabolic subalgebra} Let \(\Sigma\) be a basis for \(\Delta\). The subalgebra \(\mathfrak{b} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta^+} \mathfrak{g}_\alpha\) is @@ -601,13 +638,14 @@ This is entirely analogous to the situation of \(\mathfrak{sl}_3(K)\), where we found that the weights of the simple \(\mathfrak{sl}_3(K)\)-modules formed continuous strings symmetric with respect to the lines \(K \alpha\) with \(\kappa(\epsilon_i - \epsilon_j, \alpha) = 0\). As in the case of -\(\mathfrak{sl}_3(K)\), the same class of arguments leads us to the +\(\mathfrak{sl}_3(K)\), the same sort of arguments leads us to the conclusion\dots \begin{definition}\index{Weyl group} - We refer to the group \(W = \langle \sigma_\alpha : \alpha \in - \Delta^+ \rangle \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl - group of \(\mathfrak{g}\)}. + We refer to the (finite) group \(W = \langle \sigma_\alpha : \alpha \in + \Delta \rangle = \langle \sigma_\beta : \beta \in \Sigma \rangle \subset + \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl group of + \(\mathfrak{g}\)}. \end{definition} \begin{theorem}\label{thm:irr-weight-class} @@ -617,44 +655,79 @@ conclusion\dots the orbit of \(\lambda\) under the action of the Weyl group \(W\). \end{theorem} -\index{Weyl group!actions} +At this point we are basically done with results regarding the geometry of the +weights of \(M\), but it is convenient to introduce some further notation. Aside from showing up in the previous theorem, the Weyl group will also play an important role in chapter~\ref{ch:mathieu} by virtue of the existence of a -canonical action of \(W\) on \(\mathfrak{h}\). By its very nature, -\(W\) acts in \(\mathfrak{h}^*\). If we conjugate the action -\(\sigma\!\restriction_{\mathfrak{h}^*} : \mathfrak{h}^* \isoto -\mathfrak{h}^*\) of some \(\sigma \in W\) by the isomorphism -\(\mathfrak{h}^* \isoto \mathfrak{h}\) afforded by the restriction of the -Killing for to \(\mathfrak{h}\) we get a linear automorphism -\(\sigma\!\restriction_{\mathfrak{h}} : \mathfrak{h} \isoto \mathfrak{h}\). As -it turns out, the \(\sigma\!\restriction_{\mathfrak{h}}\) can be extended to an -automorphism of Lie algebras \(\mathfrak{g} \isoto \mathfrak{g}\). This -translates into the following results, which we do not prove -- but see +canonical action of \(W\) on \(\mathfrak{h}\). + +\begin{definition}\index{Weyl group!natural action}\index{Weyl group!dot action} + The canonical action of \(W\) on \(\mathfrak{h}^*\) given by \(\sigma \cdot + \lambda = \sigma(\lambda)\) is called \emph{the natural action of \(W\)}. We + also consider the equivalent ``shifted'' action \(\sigma \bullet \lambda = + \sigma(\lambda + \rho) - \rho\) of \(W\) on \(\mathfrak{h}^*\), known as + \emph{the dot action of \(W\)} -- here \(\rho = \sfrac{1}{2} \beta_1 + + \cdots \sfrac{1}{2} \beta_r\). +\end{definition} + +This already allow us to compute some examples of Weyl groups. + +\begin{example} + Suppose \(\mathfrak{g} = \mathfrak{sl}_n(K)\) and \(\mathfrak{h} \subset + \mathfrak{g}\) is as in Example~\ref{ex:cartan-of-sl}. Let \(\epsilon_1, + \ldots, \epsilon_n \in \mathfrak{h}^*\) be as in + Example~\ref{ex:sl-canonical-basis} and take the associated basis \(\Sigma = + \{\beta_1, \ldots, \beta_{n-1}\}\) for \(\Delta\), \(\beta_i = \epsilon_i - + \epsilon_{i + 1}\). Then a simple calculation shows that \(\sigma_{\beta_i}\) + permutes \(\epsilon_i\) and \(\epsilon_{i+1}\) and fixes the other + \(\epsilon_j\). This translates to a canonical isomorphism + \begin{align*} + W & \isoto S_n \\ + \sigma_{\beta_i} & \mapsto \sigma_i = (i \; i\!+\!1) + \end{align*} +\end{example} + +\begin{example} + 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 + Example~\ref{ex:sp-canonical-basis} and take the associated basis \(\Sigma = + \{\beta_1, \ldots, \beta_n\}\) for \(\Delta\). Then a simple calculation + shows that \(\sigma_{\beta_i}\) permutes \(\epsilon_i\) and + \(\epsilon_{i+1}\) 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) + \end{align*} +\end{example} + +If we conjugate some \(\sigma \in W\) by the isomorphism \(\mathfrak{h}^* +\isoto \mathfrak{h}\) afforded by the restriction of the Killing for to +\(\mathfrak{h}\) we get a linear action of \(W\) on \(\mathfrak{h}\), which is +given by \(\kappa(\sigma \cdot H, \cdot) = \sigma \cdot \kappa(H, \cdot)\). As +it turns out, this action can be extended to an action of \(W\) on +\(\mathfrak{g}\) by automorphisms of Lie algebras. This translates into the +following results, which we do not prove -- but see \cite[sec.~14.3]{humphreys}. \begin{proposition}\label{thm:weyl-group-action} - Given \(\alpha \in \Delta^+\), let\footnote{Notice that since $\mathfrak{g}$ - is finite-dimensional, $\operatorname{ad}(X)$ is nilpotent for each root - vector $X \in \mathfrak{g}$, so that the linear automorphism - $e^{\operatorname{ad}(X)} = \operatorname{Id} + \operatorname{ad}(X) + - \frac{\operatorname{ad}(X)^2}{2!} + \cdots : \mathfrak{g} \isoto - \mathfrak{g}$ is well defined.} \(\tilde{\sigma}_\alpha = - e^{\operatorname{ad}(E_\alpha)} e^{- \operatorname{ad}(F_\alpha)} - e^{\operatorname{ad}(E_\alpha)} : \mathfrak{g} \isoto \mathfrak{g}\). Then - \(\tilde\sigma_\alpha\) is an automorphism of Lie algebras, and this defines - an action of \(W\) on \(\mathfrak{g}\) which is compatible with the - canonical action of \(W\) on \(\mathfrak{h}\) -- i.e. - \(\tilde\sigma\!\restriction_{\mathfrak{h}} = - \sigma\!\restriction_{\mathfrak{h}}\) for all \(\sigma \in W\). + Given \(\alpha \in \Delta^+\), there is an automorphism of Lie algebras + \(f_\alpha : \mathfrak{g} \isoto \mathfrak{g}\) such that + \(f_\alpha(H) = \sigma_\alpha \cdot H\) for all \(H \in \mathfrak{h}\). In + addition, these automorphisms can be chosen in such a way that the family + \(\{f_\alpha\}_{\alpha \in \Delta^+}\) defines an action of \(W\) on + \(\mathfrak{g}\) -- which is obviously compatible with the natural action of + \(W\) on \(\mathfrak{h}\). \end{proposition} \begin{note} - Notice that the action of \(W\) on \(\mathfrak{g}\) from + We should notice the action of \(W\) on \(\mathfrak{g}\) from Proposition~\ref{thm:weyl-group-action} is not canonical, since it depends on - the choice of \(E_\alpha\) and \(F_\alpha\). Nevertheless, \(\mathfrak{h}\) - is stable under the action of \(W\) -- i.e. \(W \cdot - \mathfrak{h} \subset \mathfrak{h}\) -- and the restriction of this action to - \(\mathfrak{h}\) is independent of any choices. + the choice of \(E_\alpha\) and \(F_\alpha\). Nevertheless, different choices + of \(E_\alpha\) and \(F_\alpha\) yield isomorphic actions and the restriction + of these actions to \(\mathfrak{h}\) is independent of any choices. \end{note} We should point out that the results in this section regarding the geometry
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -166,7 +166,7 @@ this last construction. \end{align*} \end{example} -\begin{example} +\begin{example}\label{ex:sp2n} The Lie algebra of the affine algebraic group \[ \operatorname{Sp}_{2 n}(K)