Abstract
<p> Given any symmetric Cartan datum, Lusztig has provided a pair of inductive key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusztig coincide. Consequently we can define two <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z 2 times upper I"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo> × </mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}_{2} \times I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -colored graphs and prove that they are isomorhic. This result finishes the proof of the statement that Lusztig’s functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper Z"> <mml:semantics> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Z</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">f_{Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of irreducible components form a basis of the enveloping algebra. We can also deduces its crystal structure from those Lusztig’s functions. As an application, we prove that the transition matrix between the canonical basis (at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v equals 1"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">v=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) and the semicanonical basis is upper triangular with diagonal entries <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p>