On the extensions of certain representations of reductive algebraic groups with Frobenius maps

Let G be a connected reductive algebraic group defined over the finite field F-q with q elements, where q is a power of a prime number p. Let k be a field and we study the extensions of certain kG-modules in this paper. We show that the extensions of any modules in O(G) by a finite-dimensional kG-module is zero if chark >= 0 and chark not equal 2,3,p, where O(G) is the principal representation category defined in [8]. We determine the necessary and sufficient condition for the vanishing of extensions between naive induced modules. As an application, we give the conditions for the vanishing of extensions between simple modules in O(G) for G=SL2(F<overline>(q)). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.