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We study walk algebras and Hecke algebras for Kac-Moody root systems. Eachchoice of orientation for the set of real roots gives rise to a corresponding"oriented" basis for each of these algebras. We show that the notion ofdistinguished subexpression naturally arises when studying the transitionmatrix between oriented bases. We then relate these notions to the geometry ofKac-Moody flag varieties and Bott-Samelson varieties. In particular, we showthat the number of points over a finite field in certain intersections of thesevarieties is given by change of basis coefficients between oriented bases ofthe Hecke algebra. Using these results we give streamlined derivations ofDeodhar's formula for $R$-polynomials and point-counting formulas forspecializations of nonsymmetric Macdonald polynomials $E_\lambda(\mathsf{q},t)$at $\mathsf{q}=0,\infty$.

Walk algebras, distinguished subexpressions, and point counting in Kac-Moody flag varieties