A darker color would best coordinate and offer the other end of the color spectrum. Maybe a brown or black would compliment so that you have an offset of color and it stands out.
Any hue in the yellow family will compliment any color in the violet family. Choosing a cream shade such as Butter Cream or Sandstone will bring out the lilac color.
Lilac is absolutely beautiful with a green color, you can go dark green or light green for a calming room. Another color it goes well with is a bright white color.
Well tone on tone is always a good bet so why not compliment it with a slightly darker shade of purple. Also think of the shades of nature so try a medium green.
