Tokyo Journal of Mathematics

On the Extensions of Group Schemes Deforming $\mathbf{G}_a$ to $\mathbf{G}_m$

Takashi KONDO

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Abstract

For given group schemes $\mathcal{G}^{(\lambda_i)}\ (i=1,2,\ldots)$ deforming the additive group scheme $\mathbf{G}_a$ to the multiplicative group scheme $\mathbf{G}_m$, T. Sekiguchi and N. Suwa constructed extensions: $$0\rightarrow\mathcal{G}^{(\lambda_2)}\rightarrow\mathcal{E}^{(\lambda_1,\lambda_2)}\rightarrow\mathcal{G}^{(\lambda_1)}\rightarrow0,\ \ldots,\ 0\rightarrow\mathcal{G}^{(\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)}\rightarrow\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})}\rightarrow0,\ldots$$ inductively, by calculating the group of extensions $\mathrm{Ext}^1(\mathcal{E}^{(\lambda_1,\ldots,\lambda_{n-1})},\mathcal{G}^{(\lambda_n)})$. Here we treat the group $\mathrm{Ext}^1(\mathcal{G}^{(\lambda_0)},\mathcal{E}^{(\lambda_1,\ldots,\lambda_n)})$ of extensions in the case of $n=2,\ 3$. The case of $n=2$ was studied by D. Horikawa.

Article information

Source
Tokyo J. Math., Volume 33, Number 2 (2010), 283-309.

Dates
First available in Project Euclid: 31 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1296483471

Digital Object Identifier
doi:10.3836/tjm/1296483471

Mathematical Reviews number (MathSciNet)
MR2779258

Zentralblatt MATH identifier
1213.14080

Subjects
Primary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)

Citation

KONDO, Takashi. On the Extensions of Group Schemes Deforming $\mathbf{G}_a$ to $\mathbf{G}_m$. Tokyo J. Math. 33 (2010), no. 2, 283--309. doi:10.3836/tjm/1296483471. https://projecteuclid.org/euclid.tjm/1296483471


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References

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