Modular representations of the supergroup \(Q(n)\). I.

*(English)*Zbl 1027.17004The authors study the representation theory of the algebraic supergroup \(Q(n)\) in positive characteristic \(p\). The study starts with a construction of a special superalgebra, \(\text{Dist}(G)\), of distributions on \(G\). It is shown that there is an explicit equivalence between the category of representations of \(G\) and the category of “integrable” \(\text{Dist}(G)\)-supermodules.

Further, it is shown that the irreducible representations of \(G\) can be characterized in terms of highest weights. They turn out to be parameterized by the set \[ X_p^+(n)=\left\{ (\lambda_1,\dots,\lambda_n)\in\mathbb{Z}^n : \lambda_1\geq\dots\geq \lambda_n \text{ with } \lambda_i=\lambda_{i+1} \text{ only if } p|\lambda_i \right\}. \] For \(\lambda\in X_p^+(n)\) the corresponding irreducible representation \(L(\lambda)\) is the simple socle of the induced module \(H^0(\lambda)=\text{ind}_B^G\mathfrak{u}(\lambda)\), where \(B\) is a Borel subgroup of \(G\) and \(\mathfrak{u}(\lambda)\) is a certain irreducible representation of \(B\), whose dimension is a power of \(2\). It is shown that, unlike the classical situation of algebraic groups, the self-extensions between simple modules are possible, arising from the fact that representations of the diagonal subgroup, which plays the role of the maximal torus, are not completely reducible.

The authors also prove a linkage principle involving the notion of residue content of a weight \(\lambda\in X_p^+(n)\). In particular, it follows that for any two irreducible representations whose parameters have different contents, the first extension space between these two representations is zero.

Further, the authors prove an analogue of the Steinberg tensor product theorem, showing that any irreducible representation of \(Q(n)\) decomposes into a tensor product of a restricted irreducible representation and one obtained by a Frobenius twist. The restricted irreducible representations of \(Q(n)\) are exactly those which remain irreducible over the Lie superalgebra of \(Q(n)\). Their parameters are explicitly described.

Finally, the authors discuss the polynomial representations of \(Q(n)\). They show that a representation of \(Q(n)\) is polynomial if and only if it is polynomial over the diagonal subgroup. For example, the induced representations \(H^0(\lambda)\) are polynomial whenever all \(\lambda_i\) are nonnegative. It is remarked that in positive characteristic the category of polynomial representations is just as hard to understand as the category of all rational representations. In particular, in positive characteristic, \(Q(n)\) has many one-dimensional representations, unlike \(\mathbb{C}\) where such representation is unique.

Further, it is shown that the irreducible representations of \(G\) can be characterized in terms of highest weights. They turn out to be parameterized by the set \[ X_p^+(n)=\left\{ (\lambda_1,\dots,\lambda_n)\in\mathbb{Z}^n : \lambda_1\geq\dots\geq \lambda_n \text{ with } \lambda_i=\lambda_{i+1} \text{ only if } p|\lambda_i \right\}. \] For \(\lambda\in X_p^+(n)\) the corresponding irreducible representation \(L(\lambda)\) is the simple socle of the induced module \(H^0(\lambda)=\text{ind}_B^G\mathfrak{u}(\lambda)\), where \(B\) is a Borel subgroup of \(G\) and \(\mathfrak{u}(\lambda)\) is a certain irreducible representation of \(B\), whose dimension is a power of \(2\). It is shown that, unlike the classical situation of algebraic groups, the self-extensions between simple modules are possible, arising from the fact that representations of the diagonal subgroup, which plays the role of the maximal torus, are not completely reducible.

The authors also prove a linkage principle involving the notion of residue content of a weight \(\lambda\in X_p^+(n)\). In particular, it follows that for any two irreducible representations whose parameters have different contents, the first extension space between these two representations is zero.

Further, the authors prove an analogue of the Steinberg tensor product theorem, showing that any irreducible representation of \(Q(n)\) decomposes into a tensor product of a restricted irreducible representation and one obtained by a Frobenius twist. The restricted irreducible representations of \(Q(n)\) are exactly those which remain irreducible over the Lie superalgebra of \(Q(n)\). Their parameters are explicitly described.

Finally, the authors discuss the polynomial representations of \(Q(n)\). They show that a representation of \(Q(n)\) is polynomial if and only if it is polynomial over the diagonal subgroup. For example, the induced representations \(H^0(\lambda)\) are polynomial whenever all \(\lambda_i\) are nonnegative. It is remarked that in positive characteristic the category of polynomial representations is just as hard to understand as the category of all rational representations. In particular, in positive characteristic, \(Q(n)\) has many one-dimensional representations, unlike \(\mathbb{C}\) where such representation is unique.

Reviewer: Volodymyr Mazorchuk (Uppsala)

##### MSC:

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

20G99 | Linear algebraic groups and related topics |

##### Keywords:

supergroup; representation; category; highest weight; extension; irreducible representation
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\textit{J. Brundan} and \textit{A. Kleshchev}, J. Algebra 260, No. 1, 64--98 (2003; Zbl 1027.17004)

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