We compute <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{Ext}^{1}_B(\chi _1,\chi _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>Ext</mtext> <mi>B</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> between two characters <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\chi _1,\chi _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> of a Borel subgroup B of a split reductive group G over a finite field <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {F}_q,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and make an application to the calculation of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{Ext}^1_G(\pi _1,\pi _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>Ext</mtext> <mi>G</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>π</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> between principal series representations <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\pi _1,\pi _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>π</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G(\mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

On extensions of principal series representations

期刊论文
发表日期：

2026年1月6日

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Gautam H. Borisagar,
Asfak Soneji

摘要

We compute $\textrm{Ext}^{1}_B(\chi _1,\chi _2)$ ${Ext}_{B}^{1} (χ_{1}, χ_{2})$ between two characters $\chi _1,\chi _2$ $χ_{1}, χ_{2}$ of a Borel subgroup B of a split reductive group G over a finite field $\mathbb {F}_q,$ $F_{q},$ and make an application to the calculation of $\textrm{Ext}^1_G(\pi _1,\pi _2)$ ${Ext}_{G}^{1} (π_{1}, π_{2})$ between principal series representations $\pi _1,\pi _2$ $π_{1}, π_{2}$ of $G(\mathbb {F}_q)$ $G (F_{q})$ .