the a-function on a coxeter group w is a function a : w $\rightarrow$ n defined by lusztig which is intimately related to the partition of w into kazhdan--lusztig cells and to the representation theory of the hecke algebra of w. it is known that the identity element of w is the only element with a-value 0, while a non-identity element has a-value 1 if and only if it has a unique reduced word. however, as its definition relies on the kazhdan--lusztig basis of the hecke algebra, thea-function is often difficult to compute for general elements. in this talk we will focus on elements of a-value 2, or a-2 elements. we show that a-2 elements arefully commutative in the sense of stembridge, which allows us to associate to them certain posets called heaps and, in many cases, certain generalized temperley--lieb diagrams. using heaps and temperley--lieb diagrams, we conjecture a combinatorial characterization of a-2 elements, classify all coxeter groups with finitely many a-2 elements, and enumerate a-2 elements for all groups from the classification. joint with richard green.